HW 6.9.2: Complex Numbers in Polar Form and

Euler Numbers Rewrite each complex number from polar form into a bi+ form.

1. 3e5π6i 2. 3e

π2i 3. 66

ieπ

4. 7e4π3i 5.

543i

eπ

6. 6e5π4i

Rewrite each complex number into polar ire θ form. 7. 6 8. −2 9. 4i− 10. 3i

11. 2 2i+ 12. 2 2 + 2 2i 13. −3 2 + 3 2i 14. −2 + 3i

15. 5 3 − 5i 16. −3− 3 3i 17. 2 3 + 2i 18. − 3 − i

Euler’s Formula z = reiθ = r cos(θ )+ ir sin(θ )

Selected Answers:

1. 3e5π6i= 3cos 5π

6⎛⎝⎜

⎞⎠⎟ + i3sin

5π6

⎛⎝⎜

⎞⎠⎟ = − 3 3

2+ 32i

3. 6𝑒!!! = 6cos !

! + i6sin !

! = 6 !

!+ 𝑖6 !

! = 3 3 + 3i

5. 3𝑒!!! ! = 3cos !!

! + i3sin !!

! = 3 − !

!+ 𝑖3 − !

! = − ! !

! − ! !

!𝑖

7. 6 = x +yi so x = 6 and y = 0. Also, r2 = x2 + y2 = 62 + 02 = 62 so r = 6 (since r ≥ 0). Also, using x =

rcos(θ): 6 = 6cos(θ), so θ = 0. So the polar form is 6e0i.

9. -4i = x +yi so x = 0 and y = -4. Also, r2 = x2 + y2 = 02 + (-4)2 = 16 so r = 4 (since r ≥ 0). Using y =

rsin(θ): -4 = 4 sin(θ), so sin(θ) = -1, so θ = !!!

. So the polar form is 4𝑒!!! !.

11. 2 + 2i = x +yi so x = y = 2. Then r2 = x2 + y2 = 22 + 22 = 8, so r = 2 2 (since r ≥ 0). Also x =

rcos(θ) and y = rsin(θ), so:

2 = 2 2 cos(θ) so cos(θ) = !! = !

!, and 2 = 2 2 sin(θ) so sin(θ) = !

! = !

!

Therefore (x, y) is located in the quadrant I, or θ = !!. So the polar form is 2 2𝑒

!!!.

13. 6𝑒!!! !.

15. 10e11π6i

17. 4eπ6i

Notes: Polar Form of Complex Numbers Remember, because the complex plane is analogous to the Cartesian plane that we can think of a complex number z x yi= + as analogous to the Cartesian point (x, y) and recall how we converted from (x, y) to polar (r, θ) coordinates in the last section. Bringing in all of our old rules we remember the following:

rx=)cos(θ )cos(θrx =

ry=)sin(θ )sin(θry =

xy=)tan(θ 222 ryx =+

With this in mind, we can write cos( ) sin( )z x yi r irθ θ= + = + . Example 6

Express the complex number i4 using polar coordinates. On the complex plane, the number 4i is a distance of 4 from the origin

at an angle of 2π , so ⎟

⎠⎞⎜

⎝⎛+⎟

⎠⎞⎜

⎝⎛=

2sin4

2cos44 ππ ii

Note that the real part of this complex number is 0.

In the 18th century, Leonhard Euler demonstrated a relationship between exponential and trigonometric functions that allows the use of complex numbers to greatly simplify some trigonometric calculations. While the proof is beyond the scope of this class, you will likely see it in a later calculus class. Polar Form of a Complex Number and Euler’s Formula

The polar form of a complex number is z = reiθ , where Euler’s Formula holds: z = reiθ = r cos(θ )+ ir sin(θ ) Similar to plotting a point in the polar coordinate system we need r and θ to find the polar form of a complex number.

Example 7

Find the polar form of the complex number -7.

x + yi

r

θ y

x real

imaginary

4i

4

Treating this as a complex number, we can consider the unsimplified version -7+0i. Plotted in the complex plane, the number -7 is on the negative horizontal axis, a distance of 7 from the origin at an angle of π from the positive horizontal axis. The polar form of the number -7 is πie7 . Plugging r = 7 and θ = π back into Euler’s formula, we have:

707)sin(7)cos(77 −=+−=+= iiei πππ as desired. Example 8

Find the polar form of i44 +− . On the complex plane, this complex number would correspond to the point (-4, 4) on a Cartesian plane. We can find the distance r and angle θ as we did in the last section.

222 yxr += 222 4)4( +−=r 2432 ==r

To find θ, we can use rx=)cos(θ

22

244)cos( −=−=θ

This is one of known cosine values, and since the point is in the second

quadrant, we can conclude that 43πθ = .

The polar form of this complex number is i

e 43

24π

.

We can also use xy=)tan(θ to find the angle, but we must remember to check the quadrant.

Try it Now

3. Write 1+ 2i in polar form. Example 9

Write 2e2π3i in complex a bi+ form.

2e2π3i= 2cos 2π

3⎛⎝⎜

⎞⎠⎟ + i2sin

2π3

⎛⎝⎜

⎞⎠⎟ Evaluate the trig functions

-4+4i

= 2 − 12

⎛⎝⎜

⎞⎠⎟ + i2

32

⎛

⎝⎜⎞

⎠⎟ Simplify

= −1+ i 3 The polar form of a complex number provides a powerful way to compute powers and roots of complex numbers by using exponent rules from algebra. To compute a power of a complex number:

1) Convert to polar form 2) Raise to the power, using exponent rules to simplify 3) Convert back to a + bi form, if needed