Calculus questions on trigonometric functions

Anonymous
timer Asked: Dec 3rd, 2017
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Question description

For a positive integer

n, the complex number z = r(cos θ + i sin θ) has exactly n distinct nth roots given by

nr
cos
θ + 2πk
n
+ i sin
θ + 2πk
n
where k = 0, 1, 2, . . . , n − 1.

Consider the following.

Cube roots of −4

2
(−1 + i)

(a) Use the formula to find the indicated roots of the complex number. (Enter your answers in trigonometric form.)

k = 0:
k = 1:

k = 2:


(b) Represent each of the roots graphically.


(c) Write each of the roots in standard form. (Round your answers to four decimal places.)
k = 0:

k = 1:

k = 2:

2./5.55 pointsLarPreCalcRMRPAlt6 7.5.068.My Notes

Question Part
Points

Perform the operation and leave the result in trigonometric form.

3
2
cos
π
6
+ i sin
π
6
6
cos
π
4
+ i sin
π
4

3./5.55 pointsLarPreCalcRMRPAlt6 7.5.076.My Notes

Question Part
Points

Perform the operation and leave the result in trigonometric form.

cos
5π
4
+ i sin
5π
4
cos π + i sin π

4./5.55 pointsLarPreCalcRMRPAlt6 7.5.078.My Notes

Question Part
Points

Perform the operation and leave the result in trigonometric form.

4(cos 60° + i sin 60°)
5(cos 100° + i sin 100°)

5./5.55 pointsLarPreCalcRMRPAlt6 7.5.074.My Notes

Question Part
Points

Perform the operation and leave the result in trigonometric form. (Let 0 ≤ θ < 2π.)

3(cos(3.6) + i sin(3.6))
2(cos(1.4) + i sin(1.4))

6./5.55 pointsLarPreCalcRMRPAlt6 7.5.070.MI.My Notes

Question Part
Points

Perform the operation and leave the result in trigonometric form. (Enter your angles in degrees between 0° and 360°.)

1
3
cos 130° + i sin 130°
4
5
cos 300° + i sin 300°


7./5.55 pointsLarPreCalcRMRPAlt6 7.5.048.MI.My Notes

Question Part
Points

Find the standard form of the complex number. Then represent the complex number graphically.5(cos 45° + i sin 45°)



8./5.55 pointsLarPreCalcRMRPAlt6 7.5.034.My Notes

Question Part
Points

Represent the complex number graphically.

7
2
(
3
i)


Write the trigonometric form of the number.

(Let 0 ≤ θ < 2π.)


z =

9./5.55 pointsLarPreCalcRMRPAlt6 7.5.110.My Notes

Question Part
Points

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.(3 − 2i)8

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G-Mind
School: Carnegie Mellon University

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Anonymous
Outstanding Job!!!!

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