im stuck on these questions.......
6. Change -4√2 - 4√2 to trigonometric form
A. 32 cis 135°
B. 8 cis 225°
C. 8 cis 45°
D. 32 cis 45°
7. Simplify (2 cis 100°)7.
A. 2 cis 700°
B. 128 cis 280°
C. 128 cis 340°
D. 2 cis 340°
9. Simplify
12(cos52° + i sin52°)/8(cos128°+isin128°)
16. 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)
17. 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
18. Simplify (4 − 9i ) − (2 − 4i ).
A. 2 − 13i
B. 2 − 5i
C. −5 + 2i
D. 2 + 13i
6. since both x and y are negative, we are in QIII, so (C). Luckily r=8 fits as well.
7. (C): 2^7 cis(7*100) = 128 cis 340
9. 12/8 cis(52-128) = 3/2 cis -76
16. (B)
17. (D)
18. (B)
Let's go through each of these questions step by step and explain how to arrive at the answers.
6. To change -4√2 - 4√2 to trigonometric form, we need to find the magnitude and angle of the complex number. The magnitude (r) can be found using the Pythagorean theorem: r = sqrt((-4√2)^2 + (-4√2)^2) = sqrt(32 + 32) = sqrt(64) = 8.
The angle (θ) can be found using the inverse tangent function: θ = tan^(-1)((-4√2) / (-4√2)) = tan^(-1)(1) = 45°.
So the trigonometric form of the complex number is 8 cis 45°. Therefore, the answer is D.
7. To simplify (2 cis 100°)7, we need to raise the complex number to the power of 7. This can be done by multiplying the magnitudes and adding the angles: (2 cis 100°)^7 = 2^7 cis (100° * 7) = 128 cis 700°.
Therefore, the answer is A.
9. To simplify (12(cos52° + i sin52°))/[(8(cos128°+isin128°))], we can divide the magnitudes and subtract the angles:
(12/8) cis (52° - 128°) = 1.5 cis (-76°)
Therefore, the simplified form is 1.5 cis (-76°).
16. Let's analyze each statement:
(i) r = 4 – 3 sin θ is the equation for a limaçon rotated 90° - This statement is false. A limaçon rotated 90° would have an equation of r = 4 – 3 cos θ.
(ii) r = 3 cos 8θ is the equation for a rose curve with 8 petals - This statement is true. A rose curve with 8 petals has the equation r = a cos (nθ), where a is the amplitude and n is the number of petals. In this case, a = 3 and n = 8.
(iii) rθ = 16 is the equation for a circle - This statement is false. The equation for a circle is r = constant. In this case, the equation represents a line with a constant slope.
Therefore, the answer is B, (i) only.
17. To 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, we need to determine the difference in angles between the two equations: θ - (θ - 30°) = 30°.
So the rotation required is 30° clockwise. Therefore, the answer is D.
18. To simplify (4 − 9i ) − (2 − 4i ), we can subtract the real and imaginary parts individually: (4 - 2) - (9 - (-4))i = 2 - 5i.
Therefore, the answer is B, 2 - 5i.
I hope this helps you understand how to approach these questions and arrive at the correct answers!