A Graphical Approach to Precalculus with Limits (7th Edition)

Chapter 11

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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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; 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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; 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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; Express the complex number in the ... more

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; Express the complex number in the ... more

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; Express the complex number in the ... 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 in the ... more

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; Express the complex number in the ... more

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; Express the complex number in the ... more

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; 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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; ; ; ; Express the given complex number or in ... more

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; ; ; ; Express the given complex number in the ... more

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; ; ; ; Express the given complex number in the ... more

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; ; ; ; Express the given complex number in the ... more

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; ; ; ; Express the given complex number in the ... 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 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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; 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 ; 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 ... 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 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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; 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 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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