2. Choose the point that lies on the curve r = 2 – 3 sin θ.
A. (-5, 3π/2)
B. (–2, π)
C. (1, π/2)
D. (5, π/2
3. Which of the following is not an approximate solution of x5 – 1 = 0?
A. 0.3090 – 0.9511i
B. 0.8090 + 0.5878i
C. 0.3090 + 0.9511i
D. –0.8090 + 0.5878i
5. Change -4√2 - 4√2i to trigonometric form.
A. 32 cis 135°
B. 8 cis 225°
C. 8 cis 45°
D. 32 cis 45°
6. Simplify (2 cis 100°)7.
A. 2 cis 700°
B. 128 cis 280°
C. 128 cis 340°
D. 2 cis 340°
8. Simplify 12(cos 52° + i sin 52°)/ 8(cos 128° + i sin 128°)
A. 3/2cis 152°
B. 3/2cis 76°
C. 3/2cis 180°
D. 3/2cis 284°
9. Simplify i 45.
A. –i
B. 1
C. i
D. –1
10. Given the rectangular-form point (–1, 4), which of the following is an approximate primary representation in polar form?
A. (4.12, 1.82)
B. −(4.12, 1.82)
C. (−4.12, −1.33)
D. (4.12, 4.96)
11. Simplify (√2 cis 47°)(3√8
A. 48 cis 223°
B. 12 cis 223°
C. 48 cis 136°
D. 12 cis 136°
12. Which of the following statements are true?
(i) r = 4 – 3 sin θ is the equation for a limaçon rotated 90°.
(ii) r = 3 cos 8θ is the equation for a rose curve with 8 petals.
(iii) rθ = 16 is the equation for a circle.
A. (ii) only
B. (i) only
C. (i) and (ii)
D. (i) and (iii)
13. Describe the rotation required to transform the graph of r = 4 − cos (θ − 30°) sin (θ − 30°) to the graph of r = 4 − cos θ sin θ.
A. 30° counterclockwise
B. 60° clockwise
C. 60° counterclockwise
D. 30° clockwise
14. Simplify (4 − 9i ) − (2 − 4i ).
A. 2 − 13i
B. 2 − 5i
C. −5 + 2i
D. 2 + 13i
15. Find the absolute value of –2 + 6i.
A. 4
B. 4√2
C. 2√2
D. 2√10
if you're going to do a homework dump, it's better if you show your work and where you get stuck.
2. To identify the point that lies on the curve r = 2 - 3sinθ, we need to substitute the given values of (r, θ) into the equation and see if it satisfies the equation. Let's check the options:
A. (-5, 3π/2): Substitute r = -5 and θ = 3π/2 into the equation: -5 = 2 - 3sin(3π/2). This is not satisfied, so option A is not correct.
B. (–2, π): Substitute r = -2 and θ = π into the equation: -2 = 2 - 3sin(π). This is not satisfied, so option B is not correct.
C. (1, π/2): Substitute r = 1 and θ = π/2 into the equation: 1 = 2 - 3sin(π/2). This is not satisfied, so option C is not correct.
D. (5, π/2): Substitute r = 5 and θ = π/2 into the equation: 5 = 2 - 3sin(π/2). This is satisfied, so option D is correct.
Therefore, the point that lies on the curve r = 2 - 3sinθ is (5, π/2).
3. To find the approximate solutions of x^5 - 1 = 0, we need to solve this equation. However, it is not explicitly mentioned how to solve it. One approach to approximate solutions is by using numerical methods, such as Newton's method or the bisection method. Since the options provided seem to already be approximate solutions, we can check which option does not satisfy the equation:
A. 0.3090 - 0.9511i: Substitute this complex number into the equation: (0.3090 - 0.9511i)^5 - 1 = 0. This is not satisfied.
B. 0.8090 + 0.5878i: Substitute this complex number into the equation: (0.8090 + 0.5878i)^5 - 1 = 0. This is not satisfied.
C. 0.3090 + 0.9511i: Substitute this complex number into the equation: (0.3090 + 0.9511i)^5 - 1 = 0. This is not satisfied.
D. -0.8090 + 0.5878i: Substitute this complex number into the equation: (-0.8090 + 0.5878i)^5 - 1 = 0. This is satisfied.
Therefore, option D is not an approximate solution of the equation x^5 - 1 = 0.
5. To change -4√2 - 4√2i to trigonometric form, we first need to convert it to the form a + bi, where a and b are real numbers. Let's simplify the given expression:
-4√2 - 4√2i = -4(√2 + √2i) = -4(√2(√2 + i)) = -8√2 - 8√2i.
Now, let's express this in terms of the trigonometric form, r cis θ, where r is the magnitude and θ is the argument:
Magnitude (r) = √((-8√2)^2 + (-8√2)^2) = √(128 + 128) = √256 = 16.
Argument (θ) = arctan((-8√2i)/(-8√2)) = arctan(i) = π/4 + πk, where k is an integer.
Since the argument is π/4 + πk, we need to find which quadrant the complex number lies in. In this case, it lies in the third quadrant.
Therefore, the trigonometric form of -4√2 - 4√2i is 16 cis (π + π/4) = 16 cis (5π/4).
Thus, the answer is option A: 32 cis 135°.
Feel free to ask about other questions as well!