QUESTIONS IM STUCK ON

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. To change -4√2 - 4√2 to trigonometric form, we can use Euler's formula, which states that e^(ix) = cos(x) + isin(x).

The given expression, -4√2 - 4√2, can be written as -4√2(1 + i).

Now, let's convert this expression to trigonometric form:

We know that √2 = 2^(1/2), so we can rewrite it as:

-4 * 2^(1/2)(1 + i)

Next, we can express 2^(1/2) in trigonometric form:

2^(1/2) = 2^(1/2) * (cos(45°) + isin(45°))

Now, let's substitute this into our expression:

-4 * 2^(1/2)(1 + i) = -4 * (2^(1/2))(cos(45°) + isin(45°))(1 + i)

Expanding this expression, we get:

-4 * (2^(1/2))(cos(45°) + isin(45°))(1 + i) = -4 * (2^(1/2))(cos(45°) + isin(45°)) + (-4 * 2^(1/2))(cos(45°) + isin(45°))(i)

Using the distributive property, we can simplify this further:

-4 * (2^(1/2))(cos(45°) + isin(45°)) + (-4 * 2^(1/2))(cos(45°) + isin(45°))(i) = -4 * (2^(1/2))(cos(45°) + isin(45°)) - 4 * (2^(1/2))(cos(45°) + isin(45°))i

Finally, we can simplify the trigonometric form:

-4 * (2^(1/2))(cos(45°) + isin(45°)) - 4 * (2^(1/2))(cos(45°) + isin(45°))i = -8√2(cos(45°) + isin(45°))

Therefore, the trigonometric form of -4√2 - 4√2 is -8√2(cos(45°) + isin(45°)).

7. To simplify (2 cis 100°)^(7), we can use De Moivre's Theorem, which states that (cisθ)^n = cis(nθ).

Given that we have (2 cis 100°)^(7), we can apply De Moivre's Theorem:

(2 cis 100°)^(7) = 2^7 cis(7 * 100°)

Simplifying further, we get:

2^7 cis(7 * 100°) = 128 cis 700°

Therefore, the simplified form of (2 cis 100°)^(7) is 128 cis 700°.

9. To simplify (12(cos52° + i sin52°))/(8(cos128° + i sin128°)), we can divide the real parts and imaginary parts separately.

Dividing the real parts:
12(cos52°)/8(cos128°) = (12/8)(cos(52° - 128°)) = (3/2)(cos(-76°)) = (3/2)(cos76°)

Dividing the imaginary parts:
12(sin52°)/8(sin128°) = (12/8)(sin(52° - 128°)) = (3/2)(sin(-76°)) = (3/2)(-sin76°)

Therefore, the simplified form is:

(3/2)(cos76°) + (3/2)(-sin76°)i

16. Let's analyze each statement:

(i) r = 4 - 3sinθ is the equation for a limaçon rotated 90°.

A limaçon is a type of curve that forms a loop, and it can be rotated by adding or subtracting an angle from the trigonometric function. However, in this case, the equation r = 4 - 3sinθ does not rotate the limaçon by 90°. Therefore, statement (i) is false.

(ii) r = 3cos8θ is the equation for a rose curve with 8 petals.

A rose curve is a polar equation that forms a symmetric pattern with multiple petals. The number of petals is determined by the coefficient of θ. In this case, r = 3cos8θ has 8 petals, so statement (ii) is true.

(iii) rθ = 16 is the equation for a circle.

The equation rθ = k, where k is a constant, represents a circle with radius r = k/θ. In this case, rθ = 16 does not have a constant radius. Therefore, statement (iii) is false.

Based on our analysis, both statement (ii) and statement (iii) are true. Therefore, the correct answer is option D. (ii) and (iii).

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 - cosθsinθ, we need to find the difference between the angles θ - 30° and θ.

By subtracting the angles, we have:

(θ - 30°) - θ = -30°

Therefore, the rotation required is 30° counterclockwise. Hence, the correct answer is option A. 30° counterclockwise.

18. To simplify (4 - 9i) - (2 - 4i), we can subtract the real parts and the imaginary parts separately.

Subtracting the real parts: 4 - 2 = 2

Subtracting the imaginary parts: (-9i) - (-4i) = -9i + 4i = -5i

Therefore, the simplified form is 2 - 5i, which corresponds to option B. 2 - 5i.