A Graphical Approach to Precalculus with Limits (7th Edition)
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Use the angle sum property of a triangle to obtain... more
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Use the angle sum property of a triangle to obtain... more
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Use the sine law to equate the ratios and and ... more
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Use the sine law to equate the ratios and and ... more
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Use the angle sum property of a triangle to obtain... more
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There is no solution for . ; The sine law states ... more
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The sine law states that the ratios of length of ... more
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The sine law states that the ratios of length of ... more
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The sine law states that the ratios of length of ... more
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The sine law states that the ratios of length of ... more
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The AAA similarity criterion states that if all ... more
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The AAA similarity criterion states that if all ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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If the solution of a triangle may lead to zero, ... more
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; The angle sum property of a triangle states ... more
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; The angle sum property of a triangle ... more
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; The angle sum property of a triangle states ... more
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4 ; The angle sum property of a triangle states ... more
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; The angle sum property of a triangle states ... more
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, , ; The law of sines states that , where , , ... more
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, , ; The law of sines states that , where , , ... more
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, , ; The law of sines states that , where , , ... more
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No such triangle ; The law of sines states that... more
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No such triangle ; The law of sines states that... more
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; The sum of the angles of a triangle is , i.... more
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; The sum of the angles of a triangle is , i.e. .... more
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; The law of sines states that , where , , are ... more
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; Draw the given triangle with different names of... more
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11.5 ; Draw the given triangle with different ... more
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; The law of sines states that , where , , are ... more
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The value of should be less than or greater than... more
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; ; The triangle is a right triangle. ; The ... more
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Substitute 26.8 for , 31.3 for , for in the law ... more
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The angle is the obtuse angle in the triangle ... more
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Increasing ; Sketch the graph of the given ... more
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If , then because the function is increasing on ... more
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; The law of sines is , where and are the sides... more
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The equation is equivalent to .The expression is... more
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The angle is the obtuse angle in the triangle ... more
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The required values of and are 4, 3 and 10 ... more
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7 ; Consider the two sides, and with the ... more
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5 ; Consider the two sides, and with the ... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides, , and of an ... more
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; Consider the two sides, and with the included... more
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; Consider the two sides, and with the included... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides of the given oblique ... more
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; Consider the two sides of the given triangle as... more
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; In the given triangle, the lengths of two sides... more
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; Consider the two sides, and with the included... more
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; In the given problem, the lengths of two sides ... more
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; In the given problem, the lengths of two sides ... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides, , and of an oblique... more
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; In the given problem, the SSS is given.The ... more
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; In the given problem, the SSS is given.The ... more
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; Consider the three sides, , and of an oblique... more
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, , ; Consider the three sides, , and of an ... more
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; Consider the two sides, and with the included... more
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; Consider the two sides, and with the included... more
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; In the given problem, the SAS is given.The ... more
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; Consider the two sides, and with the included... more
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; Consider the two sides, and with the included... more
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; Consider the two sides, and with the included... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides, , and of an oblique... more
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; Consider the included side, and two angles ... more
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; Consider the one side, and two angles and of... more
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; Consider the two sides, and with the included... more
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; Consider the two sides, and with the included... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides, , and of an oblique... more
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; Consider the two sides, and with angle of an... more
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; Consider the two sides, and with angle of an... more
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; Consider the two sides, and with angle of an... more
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Case: 2 (one triangle); , , ; Rewrite the angle ... more
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No triangle ; Rewrite as .The conversion factor ... more
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No triangle ; Rewrite as .The conversion factor ... more
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The absolute value of is 1.34 which is greater ... more
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Consider that the line is a side of a triangle... more
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; Draw a horizontal line joining the points and... more
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and ; Draw a simplified figure which is ... more
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; Draw a simplified figure which is equivalent to... more
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and ; Draw a parallelogram label its vertices as... more
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; Draw a triangle label its vertices as , , . ... more
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; Draw a triangle label its vertices as , , . ... more
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; Draw a perpendicular line at the position of ... more
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; Consider the side opposite to the angle as ... more
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; Draw an appropriate drawing depicting the given... more
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; Multiply 36.2 and 45.6 by 3 separately to ... more
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; The law of cosines states that ; ; , where... more
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Distance from the second base: ; Distance from ... more
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; Draw a parallelogram label its vertices as... more
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; Draw an appropriate drawing depicting the given... more
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; The craft S takes 2 hours to make an orbit. It ... more
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; Draw an appropriate drawing depicting the given... more
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; Draw an appropriate drawing depicting the given... more
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; Draw an appropriate diagram depicting the given... more
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; Use the law cosines to calculate the length of ... more
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; Draw an appropriate drawing depicting the given... more
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; Use the law cosines to calculate the value of .... more
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; Use the law cosines to calculate the distance ... more
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; The area formula for a triangle is , where and... more
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; The area formula for a triangle is , where and... more
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; The law of cosines states that ; ; , where... more
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; If , and are three sides of a triangle and ... more
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; If , and are three sides of a triangle and ... more
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; The semi perimeter, i.e, of the triangle is ... more
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; The semi perimeter, i.e, of the triangle is ... more
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; The semi perimeter, i.e, of the triangle is ... more
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33 object C ; Heron’s formula is given by , where... more
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; The area of the triangle is given by the ... more
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; The area of the triangle is defined by the ... more
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; Heron’s formula is given by , where , and are... more
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Substitute 9 for , 10 for and 17 for in the ... more
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Substitute 6 for , 4 for and 5 for in the ... more
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The largest angle of an obtuse triangle must range... more
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In the expression , is negative.The highest ... more
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As and , the inequalities and both are true.As... more
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The given conditions show that which violates the... more
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and ; and ; Two or more vectors are said to be ... more
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and , and , and , and ; Two vectors are said ... more
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The double of the vector is equal to the vectors... more
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The vector is twice and opposite to the vector , ... more
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; Two vectors are said to be opposite if they are... more
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; Two vectors are said to be opposite if they are... more
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; A vector is said to be a scalar multiple of ... more
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; A vector is said to be a scalar multiple of ... more
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; Draw two vectors and from the same initial ... more
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; The parallelogram rule states that place two ... more
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; The opposite of the vector is the vector such... more
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; The opposite of the vector is the vector such... more
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; The parallelogram rule states that place two ... more
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; The parallelogram rule states that place two ... more
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; The parallelogram rule states that place two ... more
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; The parallelogram rule states that place two ... more
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; The sum of two vectors and is , where and ... more
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; Evaluate the expressions .Substitute for in ... more
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; Evaluate the expressions .Substitute for in ... more
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; Substitute for in the given expression and ... more
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; Substitute for in the given expression and ... more
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; Evaluate the expressions and .Substitute for... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; In the expression , is the horizontal ... more
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; Sketch the given figure, and draw a horizontal ... more
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; Sketch the given figure, and draw a horizontal ... more
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; Sketch the given figure, and draw a horizontal ... more
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; Sketch the given figure, and draw a horizontal ... more
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; The vector can be expressed as , where is the... more
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; The vector can be expressed as , where is the... more
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; The vector can be expressed as , where is the... more
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; The vector can be expressed as , where is the... more
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; In the expression , is the horizontal ... more
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; If a vector has a magnitude and a directed ... more
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; If a vector has a magnitude and a directed ... more
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is equal to ; If a vector has a magnitude and a... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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; ; The magnitude of a vector of the form is ... more
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7 ; The dot product of the vectors and is ... more
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; The dot product of the vectors and is denoted... more
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; The dot product of the vectors and is denoted... more
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1 ; The dot product of the vectors and is ... more
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20 ; The dot product of the vectors and is ... more
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; The dot product of the vectors and is denoted... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If the angle between two vectors and is ... more
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; If , are real numbers and is a scalar, then ... more
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; If , , , are real numbers and then difference ... more
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; The dot product of the vectors and is denoted... more
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; If , are real numbers and is a scalar, then ... more
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The dot product of the vectors and is denoted by... more
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If and are two vectors, then they are orthogonal... more
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If and are two vectors, then they are orthogonal... more
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If and are two vectors, then they are orthogonal... more
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If and are two vectors, then they are orthogonal... more
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If and are two vectors, then they are orthogonal... more
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; ; Given that the angle between the first force ... more
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; ; Given that the angle between the first force ... more
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; Draw a figure representing the force of the ... more
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; Draw a figure representing the force of the car... more
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; Draw a figure representing the force of the ... more
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; Consider as the angle between first and second... more
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; Draw a figure representing the force on the ... more
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; Draw a figure representing the force on the ... more
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Weight of the crate is ; Tension on the horizontal... more
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226.23 lb ; In the given figure, represent the ... more
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; ; The bearing is angle measured clockwise from ... more
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; ; The bearing is angle measured clockwise from ... more
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; Bearing is an angle measured clockwise from the... more
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; Bearing is the angle measured clockwise from ... more
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; ; Consider as the actual speed of the boat and... more
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; The given speed of the carrier is . Multiply 32... more
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; ; Consider as the angle between the directions ... more
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; ; Consider as the angle between the directions ... more
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Airspeed=; Ground speed= ; Consider as the ground... more
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; ; Consider as the angle between the directions ... more
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; Consider as the angle between the directions ... more
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; Given that the plane has to cover 400 miles due... more
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; Draw a triangle and mark the angle as , the ... more
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Two solutions;, , ;, , ; The law of sine states ... more
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One solution;A=B=b=16.4 ; The law of sine states ... more
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9.6 ; The formula to calculate the area of the ... more
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21 ; The formula to calculate the semiperimeter of... more
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; ; The dot product rule of two vectors and ... more
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; The rectangular form of the complex number is... more
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; The rectangular form of the complex number is... more
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; The rectangular form of the complex number is... more
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; The rectangular form of the complex number is... more
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; The rectangular form of the complex number is... more
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; The rectangular form of the complex number is... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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The rectangular form of the complex number is , ... more
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magnitude. ; The modulus of the complex number is... more
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The imaginary part should be zero. ; The ... more
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Real axis. ; A complex number is graphed on a ... more
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; The sum of two (or more) complex numbers is a ... more
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; The sum of two (or more) complex numbers is a ... more
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; The sum of two (or more) complex numbers is a ... more
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; The sum of two (or more) complex numbers is a ... more
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; The sum of two (or more) complex numbers is a ... more
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; The sum of two (or more) complex numbers is a ... more
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10 ; The formula for the modulus of a complex ... more
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5 ; The formula for the modulus of a complex ... more
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13 ; The formula for the modulus of a complex ... more
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25 ; The formula for the modulus of a complex ... more
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6 ; The formula for the modulus of a complex ... more
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15 ; The formula for the modulus of a complex ... more
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; The formula for the modulus of a complex number... more
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; The formula for the modulus of a complex number... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; A complex number is said to be in the ... more
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; The formula for the modulus, of a complex ... more
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; The formula for the modulus, of a complex ... more
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; The formula for the modulus, of a complex ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; If and are two complex numbers, then the ... more
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; If and are two complex numbers then the ... more
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; If and are two complex numbers then the ... more
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; If and are two complex numbers then the ... more
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; If and are two complex numbers then the ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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; Consider the complex number in the numerator.... more
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; For a complex number of the form , the modulus ... more
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; For a complex number of the form , the modulus ... more
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The complex numbers , , , and so on, have modulus... more
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; Substitute the given values and in the given ... more
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; Substitute the given values of , , and in the... more
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; Evaluate , substitute the value of in the ... more
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The value of is equal to , similarly the value of... more
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To square a complex number in trigonometric form, ... more
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; The trigonometric form of a complex number is... more
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The imaginary parts of the two complex numbers ... more
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; The De Moivre’s theorem states that the ... more
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; The De Moivre’s theorem states that the ... more
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1 ; The De Moivre’s theorem states that the ... more
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; The De Moivre’s theorem states that the ... more
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8 ; The De Moivre’s theorem states that the ... more
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; The De Moivre’s theorem states that the ... more
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; The expression is equivalent to . Write as ... more
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; The expression is equivalent to . Write as ... more
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; Express the complex number in the ... more
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1 ; Express the complex number in the ... more
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; ; ; ; Express the complex number 1 or in the ... more
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; ; ; ; Express the complex number or in the ... more
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; ; ; ; The root theorem states that the ... more
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; ; ; ; The root theorem states that the ... more
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; ; ; ; Express the given complex number or in ... more
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; ; ; ; Express the given complex number or in ... more
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; ; ; ; Express the given complex number or in ... more
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; ; ; The root theorem states that the distinct... more
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;; ; The root theorem states that the distinct ... more
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;; ; The root theorem states that the distinct ... more
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; ; ; ; The root theorem states that the ... more
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; ; Express the given complex number in the ... more
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, , ; A complex number of the form can be written... more
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, , , ; A complex number of the form can be ... more
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, ; A complex number of the form can be written ... more
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, ; A complex number of the form can be written ... more
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For a complex number with argument and for and... more
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; The given equation can be written as .The ... more
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; The given equation can be written as .The ... more
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; The given equation can be written as .The five ... more
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; The given equation can be written as .The four ... more
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, , ; The given equation can be written as .The ... more
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; The given equation can be written as .The cube ... more
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; The given equation can be written as .The five ... more
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; A complex number of the form can be written in... more
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5 ; The modulus of a complex number is defined as... more
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; A complex number of the form can be written in... more
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; The De Moivre's theorem on complex number is ... more
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; A complex number of the form can be written in... more
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, , ; A complex number of the form can be written... more
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; The cube roots of are the solutions of the ... more
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; Substitute for in the given equation to ... more
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1 ; Substitute for in the given equation to ... more
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and ; The relationship between rectangular ... more
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; and ; The relationship between rectangular ... more
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; and ; The relationship between rectangular ... more
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; and ; Plot the given point in the graph to ... more
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; and ; Plot the given point in the graph to ... more
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and ; Plot the given point in the graph to obtain... more
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; and ; Plot the given point in the graph to ... more
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; and ; Plot the given point to obtain the ... more
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; and ; Plot the given point to obtain the ... more
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; and ; Plot the given point to obtain the ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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The relationship between rectangular coordinates ... more
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Cardioid ; Press the ON button to turn on the ... more
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cardioid ; Press the ON button to turn on the ... more
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Limacon ; Press the ON button to turn on the ... more
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Cardioid ; Press the ON button to turn on the ... more
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Four-leaved rose ; Press the ON button to turn on ... more
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Five-leaved rose ; Press the ON button to turn on ... more
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Lemniscate ; Solve the given equation for .Apply ... more
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Lemniscate ; Solve the given equation for .Apply ... more
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Cardioid ; Press the ON button to turn on the ... more
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Cardioid ; Press the ON button to turn on the ... more
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Cissoid ; Press the ON button to turn on the ... more
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Cissoid with a loop ; Press the ON button to turn ... more
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To draw the graph of the polar function , where , ... more
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Whenever the angle between two rays is , the two ... more
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The angles, must be coterminal with , , , ; ... more
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The graph of the equation of the form for is a ... more
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The graph is reflected across the line or -axis... more
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The graph of the equation is rotated in the ... more
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The graph of the equation of the form or is a ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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; ; The relationship between rectangular ... more
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Substitute 1 for in the given equation and ... more
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Substitute 2 for in the given equation and ... more
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Substitute 1.5 for in the given equation and ... more
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Substitute for in the given equation and ... more
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, ; Equate the expression of two different of the... more
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and ; Equate the expression of two different of ... more
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, ; Equate the expression of two different of the... more
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; Substitute for in the equation and simplify.... more
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; Substitute 4 for in the equation and simplify... more
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; Substitute for in the equation and simplify.... more
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, that is a circle of radius 3 and origin as ... more
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; a circle of radius 2 and center as origin. ; ... more
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; line segment ; Divide both sides of the equation... more
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; Ellipse with center ; Divide both sides of the ... more
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; upper branch of Hyperbola ; Divide both sides of... more
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; Hyperbola ; The trigonometric identities state ... more
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; It is given that the value of varies in .The ... more
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; It is given that the value of varies in .The ... more
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for x in and in ; It is given that the value of... more
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for x in and in ; It is given that the value of... more
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Substitute arbitrary values of in the first ... more
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Substitute 0 for in the given parametric ... more
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Substitute 0 for in the given parametric ... more
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Substitute 0 for in the given parametric ... more
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Substitute 0 for in the given parametric ... more
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Substitute 0 for in the given parametric ... more
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Substitute 0 for in the given parametric ... more
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Substitute 0 for in the given parametric ... more
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Letter of the alphabet graphed by the given ... more
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Letter of the alphabet graphed by the given ... more
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Letter of the alphabet graphed by the given ... more
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Letter of the alphabet graphed by the given ... more
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Turn ON the graphing calculator.Press the MODE ... more
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Turn ON the graphing calculator.Press the MODE ... more
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Turn ON the graphing calculator.Press the MODE ... more
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Turn ON the graphing calculator.Press the MODE ... more
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173 feet ; Consider a triangle ABC with sides , ... more
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147.712 centimeters ; Consider a triangle ABC ... more
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153618.673 square meters. ; For a triangle ABC, ... more
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20.26 square feet ; For a triangle ABC, the area ... more
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0.234 square kilometers ; For a triangle , the ... more
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684.631 square meters ; For a triangle ABC, the ... more
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; There are degrees in 1 minutes.Convert the ... more
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; Consider the path for the boats which starts ... more
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; The sum of the angles of the triangle is ... more
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For the given values of , and , other values of ... more
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No, the triangle is not possible because the sum ... more
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The law of cosine becomes the Pythagorean theorem... more
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For acute angle, the condition is .For obtuse ... more
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29; ; The magnitude of the vector is calculated ... more
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; The magnitude of the vector is calculated as... more
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; The position vector is expressed as , where is... more
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; The position vector is expressed as , where is... more
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; The position vector is expressed as , where is... more
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Yes, the vectors are orthogonal. The dot product ... more
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; Consider a triangle with sides , and .Side ... more
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; Consider a triangle ABC with sides AB, AC and ... more
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The complex number gives the position vector on ... more
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The complex number gives the position vector on ... more
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; Add the real and imaginary parts of the given ... more
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; Add the real and imaginary parts of the given ... more
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; The relationship among , , , and is given by... more
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; The relationship among , , , and is given by... more
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; The relationship among , , , and is given by... more
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; The relationship among , , , and is given by... more
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; The distributive property of multiplication ... more
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; The expression is equivalent to .Write the ... more
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; The quotient theorem for complex numbers states... more
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; The quotient theorem for complex numbers states... more
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; The relationship among , , , and is given by... more
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; The De Moivre's theorem states that . In ... more
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; The De Moivre's theorem states that . In ... more
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, , ; ; The relationship among , , , and is given... more
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, , , ; ; The relationship among , , , and is ... more
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, , , , ; ; The relationship among , , , and is ... more
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, , , ; ; Solve the given equation for .Subtract ... more
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; The relationship between rectangular ... more
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; The relationship between rectangular ... more
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; The relationship between rectangular ... more
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Cannot be determined ; The relationship between ... more
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Press the ON button to turn on the calculator.... more
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Press the ON button to turn on the calculator.... more
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Press the ON button to turn on the calculator.... more
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Press the ON button to turn on the calculator.... more
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; The relationship between rectangular ... more
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; The relationship between rectangular ... more
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; The relationship between rectangular ... more
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; The relationship between rectangular ... more
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; If a point has rectangular coordinate and ... more
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; If a point has rectangular coordinate and ... more
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; If a point has rectangular coordinate and ... more
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; If a point has rectangular coordinate and ... more
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; Use the trigonometric identity to eliminate ... more
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Consider some arbitrary values of in the interval... more
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; The law of sines states that in any triangle ... more
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; The law of cosines states that in any triangle ... more
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; The law of cosines states that in any triangle ... more
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168 square units ; If a triangle has sides , , ... more
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18 square units ; If two sides of a triangle are... more
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;; ; The law of cosines states that in any ... more
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, , ; , , ; The law of sines states that in any ... more
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10; ; The vector given by the diagram is .The ... more
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; ; ; ; Express the given complex number in the ... more
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; The equations for conversion of polar ... more
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Substitute arbitrary values for from the given ... more
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Substitute arbitrary values for from the given ... more
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; Use the angle sum formula to obtain the angle... more
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; Use the angle sum formula to obtain the angle... more
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; Use the angle sum formula to obtain the angle... more
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; Use the angle sum formula to obtain the angle... more
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; Use the angle sum formula to obtain the angle... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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; ; ; The angle sum property of a triangle states ... more
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2 ; The law of sines states that the ratios of the... more
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1 ; The law of sines states that the ratios of the... more
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0 ; The law of sines states that the ratios of the... more
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0 ; The law of sines states that the ratios of the... more
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Two possible triangles: , , ;, , ; The law of ... more
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Two possible triangles: , , ;, , ; The law of ... more
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No such triangle possible ; The law of sines ... more
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No such triangle possible ; The law of sines ... more
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, , ; The law of sines states that in any triangle... more
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, , ; , , ; The law of sines states that , ... more
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, , ; , , ; The law of sines states that , ... more
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, , ; , , ; The law of sines states that , ... more
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, , ; , , ; The law of sines states that , ... more
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The Pythagorean theorem can be applied in the ... more
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The given statement does not comply with the ... more
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; The line AB is exactly in the southeast ... more
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; Substitute for , for in the given expression... more
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; ; Express the given complex number in the ... more
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, ; A complex number of the form can be written ... more
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The value of determines the length of the petals ... more
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Substitute 2 for in the given equation and ... more
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1455.64 feet ; As the golf ball is on the ground, ... more
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Substitute for in the given equation for ... more
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Quadrant IV ; The formula for rectangular ... more
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, ; The formula for polar coordinates if the ... more
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Cardioid ; Since for all values of , the graph is... more
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; The relationship between rectangular ... more
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Substitute arbitrary values of in the first ... more
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; The law of sines states that the ratios of the ... more
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; The law of sines states that the ratios of the ... more
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; Consider the three sides, , and of an oblique... more
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; Consider the three sides, , and of an oblique... more
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; The law of sines states that the ratios of the ... more
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; The law of sines states that the ratios of the ... more
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1.91 miles (approximately) ; Consider the radio ... more
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; Consider as the speed of the plane and as the... more
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; Consider the resulting speed as .Subtract from... more
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358 feet ; If a ball is thrown with an initial ... more