# Trigonometry

1. Find the absolute value of the following complex number:

2. Choose the rectangular coordinates for the following polar coordinate:

3. Determine the rectangular form of the complex number:

4. Find the polar form of the following complex number:

5. When plotted on the rectangular coordinate system in which quadrant would the following point be located for this polar coordinate?

6. Find the power of the following complex number:

7. Find the value of the given complex number:

8. Choose the polar coordinates for the following rectangular coordinate

9. The following polar coordinates are multiple representations of the same point, True or False?

10. Find the polar form of the following expression:

11. Write each complex number in rectangular form. If necessary round to the nearest tenth.

12. Use DeMoirvre's Theorem to find the indicated power of the complex number. Write answer in rectangular form.

(Please see the attachment)

https://brainmass.com/math/trigonometry/trigonometry-absolute-values-complex-numbers-331426

## SOLUTION This solution is **FREE** courtesy of BrainMass!

The solution file is attached.

1. Find the absolute value of the following complex number:

a. z = 2 + 5i

|z| = âˆš(2^2 + 5^2) = âˆš29

2. Choose the rectangular coordinates for the following polar coordinate:

a. (6, )

x = r cos Î¸ = 6 cos (3 Ï€/2) = 0 and r sin Î¸ = 6 sin (3 Ï€/2) = -6

The rectangular coordinates are (x, y) = (0, -6)

3. Determine the rectangular form of the complex number:

z = 8 (cos = i sin )

cos Ï€/2 = 0 and sin Ï€/2 = 1

The rectangular form is z = 8[0 + Î¹(1)] = 8Î¹

4. Find the polar form of the following complex number:

a. z = 7 ( cos i sin )

The given complex number is already in the polar form.

5. When plotted on the rectangular coordinate system in which quadrant would the following point be located for this polar coordinate?

a. ( -2 , )

b. ( -3, )

(a) cos 2 Ï€/3 = -1/2 and sin 2 Ï€/3 = âˆš3 /2

The rectangular coordinates are (r cos Î¸, r sin Î¸) =( 2 *-1/2, 2 * âˆš3 /2) = (-1, âˆš3)

II quadrant

(b) cos Ï€/4 = 1/âˆš2 and sin Ï€/4 = 1/âˆš2

The rectangular coordinates are (r cos Î¸, r sin Î¸) =( 3 * 1/âˆš2, 3 * 1/âˆš2) = (3 âˆš2 /2, 3 âˆš2 / 2)

I quadrant

6. Find the power of the following complex number:

z = ( - i )

r cos Î¸ = âˆš2 and r sin Î¸ = -1

r^2 = (âˆš2)^2 + (-1)^2 = 3

r = âˆš3

Î¸ = 2 Ï€ + tan^-1 (y/x) = 2 Ï€ + tan^-1 (-1/âˆš2) = 9 Ï€/5

z = âˆš3 [cos(9 Ï€/5) + Î¹ sin(9 Ï€/5)] = âˆš3 [cos(-Ï€/5) Î¹ sin(-Ï€/5)]

z^4 = (âˆš3)^4 [cos(-Ï€/5) Î¹ sin(-Ï€/5)]^4 = 9 [cos (-4 Ï€/5) + Î¹ sin(-4 Ï€/5)] = 9 e^(-4 Ï€/5)Î¹

7. Find the value of the given complex number:

i

Î¹^5 = Î¹^2 * Î¹^2 * Î¹ = Î¹

8. Choose the polar coordinates for the following rectangular coordinate:

( -1, - )

r cos Î¸ = -1, r sin Î¸ = -âˆš3

r^2 = (-1)^2 + (-âˆš3)^2 = 1 + 3 = 4

r = 2

cos Î¸ = -1/2 and sin Î¸ = -âˆš3 /2

Î¸ = 4 Ï€/3

The polar coordinates are (r, Î¸) = (2, 4 Ï€/3)

9. The following polar coordinates are multiple representations of the same point, True or False?

( -5, ) ( -5, )

False, because the arguments are different in the two complex numbers

10. Find the polar form of the following expression:

3 - 3

z = 3 âˆš2 (1 - Î¹)

x = 1 = r cos Î¸ and y = -1 = r sin Î¸

r = 1^2 + (-1)^2 = 2

r = âˆš2

cos Î¸ = 1/âˆš2 and sin Î¸ = -1/âˆš2

Î¸ = 7 Ï€/4

The polar form is r (cos Î¸ + Î¹ sin Î¸) = âˆš2 [cos (7 Ï€/4) + Î¹ sin(7 Ï€/4)]

11. Write each complex number in rectangular form. If necessary round to the nearest tenth.

4 + i sin

Given Expression = 4(-0.866 + Î¹ * 0.5) = -3.5 + 2Î¹

12. Use DeMoirvre's Theorem to find the indicated power of the complex number. Write answer in rectangular form.

Given Expression = (1/2)^5 * [cos(5 * Ï€/10) + Î¹ sin(5 * Ï€/10)]

= (1/32)[cos(Ï€/2) + Î¹ sin(Ï€/2)]

= (1/32)(0 + Î¹)

= Î¹/32

https://brainmass.com/math/trigonometry/trigonometry-absolute-values-complex-numbers-331426