6.5 Trig. Form of a Complex Number

6.5 Trig. Form of a Complex Number Ex. Z = -2 + 5i 5 -2

Trigonometric Form of a Complex Number r = the hypotenuse and theta = the angle. Write in trig. form Ex. Find r (the hyp) & theta r = 4 ref. angle

Write the complex number in standard form a + bi. Ex.

Multiplication of Complex Numbers Find the Product if &

Dividing Complex Numbers

Divide

DeMoivre’s Theorem and nth Roots

Ex. Find Imagine how much fun it would be to multiply this example out 12 times. It would take forever. Using DeMovre’s Theorem, however, makes it short and simple. First, convert to trigonometric form.

nth Roots of a Complex Number For a positive integer n, the complex number z = r(cos x + i sin x) has exactly n distinct nth roots given by where k = 0, 1, 2, . . . , n - 1 WATCH THE EXAMPLES CAREFULLY!!!

Ex. Find all the sixth roots of 1. First, write 1 + 0i in trig. form. 1 + 0i is over 1 and up 0. Therefore, 1 is the hypotenuse and theta is 0o. Now, we plug in 0, 1, 2, 3, 4, and 5 for k to find our six roots.

Find the three cube roots of z = -2+2i Again, first convert to trig form. For k = 0, 1, and 2, the roots are:

6.5 Trig. Form of a Complex Number