Calculus questions on trigonometric functions

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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

 n r
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

G-Mind
School: Carnegie Mellon University

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Anonymous
Outstanding Job!!!!

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