Applications of Trigonometric Functions Chapter 7 Applications of Trigonometric Functions © 2011 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved 1

Polar Form of Complex Numbers; DeMoivre’s Theorem SECTION 7.8 Represent complex numbers geometrically. Find the absolute value of a complex number. Write a complex number in polar form. Find products and quotients of complex numbers in polar form. Use DeMoivre’s Theorem to find powers of a complex number. Use DeMoivre’s Theorem to find the nth roots of a complex number. 1 2 3 4 5 6

GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS The geometric representation of the complex number a + bi is the point P(a, b) in a rectangular coordinate system. When a rectangular coordinate system is used to represent complex numbers, the plane is called the complex plane. The x-axis is also called the real axis, because the real part of a complex number is plotted along the x-axis. Similarly, the y-axis is also called the imaginary axis. © 2011 Pearson Education, Inc. All rights reserved

GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS A complex number a + bi may be viewed as a position vector with initial point (0, 0) and terminal point (a, b). © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Plotting Complex Numbers Plot each number in the complex plane. 1 + 3i, –2 + 2i, –3, –2i, 3 – i Solution © 2011 Pearson Education, Inc. All rights reserved

ABSOLUTE VALUE OF A COMPLEX NUMBER The absolute value (or magnitude or modulus) of a complex number z = a + bi is © 2011 Pearson Education, Inc. All rights reserved

POLAR FORM OF A COMPLEX NUMBER The complex number z = a + bi can be written in polar form where a = r cos θ, b = r sin θ, and When a nonzero complex number is written in polar form, the positive number r is the modulus or absolute value of z; the angle θ is called the argument of z (written θ = arg z). © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Writing a Complex Number in Rectangular Form EXAMPLE 4 Write the complex number in rectangular form. Solution The rectangular form of z is © 2011 Pearson Education, Inc. All rights reserved

PRODUCT AND QUOTIENT RULES FOR TWO COMPLEX NUMBERS IN POLAR FORM Let z1 = r1(cos 1 + i sin 1) and z2 = r2(cos θ2 + isin θ2) be two complex numbers in polar form. Then and © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Finding the Product and Quotient of Two Complex Numbers EXAMPLE 5 Leave the answers in polar form. Solution © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved Finding the Product and Quotient of Two Complex Numbers EXAMPLE 5 Solution continued © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved DEMOIVRE’S THEOREM Let z = r(cos + i sin) be a complex number in polar form. Then for any integer n, © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Power of a Complex Number Let z = 1 + i. Use DeMoivre’s Theorem to find each power of z. Write answers in rectangular form. Solution Convert z to polar form. Find r and . , © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Power of a Complex Number Solution continued © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Power of a Complex Number Solution continued © 2011 Pearson Education, Inc. All rights reserved

DEMOIVRE’S nth ROOTS THEOREM Let z and w be two complex numbers and let n be a positive integer. The complex number z is called an nth root of w if zn = w. The nth roots of a complex number w = r(cos  + i sin ), where r > 0 and  is in degrees, are given by for k = 0, 1, 2, …, n – 1. If  is in radians, replace 360º with 2π in zk. © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Roots of a Complex Number Find the three cube roots of 1 + i in polar form, with the argument in degrees. Solution In the previous example, we showed that and . Use DeMoivre’s Theorem with n = 3. © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Roots of a Complex Number Solution continued Substitute k = 0, 1, and 2 in the expression for zk and simplify to find the three cube roots. © 2011 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Roots of a Complex Number Solution continued The three cube roots of 1 + i are as follows: © 2011 Pearson Education, Inc. All rights reserved