1. 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. (i) and (iii)
B. (i) only
C. (ii) only
D. (i) and (ii)

2) How many petals do the two rose curves r = 9 cos 61θ and r = 8 sin 60θ have combined?
A. 182
B. 242
C. 181
D. 131

18. What is true of the curve r = ƒ(θ) if its shape is unaffected when r is replaced by –r and θ is replaced by p – θ?
A. It's symmetric with respect to the x-axis.
B. It's symmetric with respect to the origin.
C. It's symmetric with respect to the line y = x.
D. It's symmetric with respect to the y-axis.

you should be able to figure these out with a visit to wolframalpha.com

Type in

plot r = 4 – 3 sin θ

to see that (i) is true

If you still have trouble, c'mon back...

1) To determine which statements are true, let's break down each equation and analyze their properties:

(i) r = 4 - 3 sin θ represents a limaçon. A limaçon is a type of polar curve that consists of a loop and a cusp. When the limaçon is rotated by 90 degrees, the loop becomes vertical and the cusp becomes horizontal. Therefore, statement (i) is true.

(ii) r = 3 cos 8θ represents a rose curve. A rose curve is a type of polar curve that exhibits symmetry in the number of petals. The number of petals can be determined by dividing the coefficient of θ by 2. In this case, 8 divided by 2 equals 4, so the rose curve has 4 petals. Therefore, statement (ii) is false.

(iii) rθ = 16 represents a circle in polar coordinates. A circle in polar coordinates has a constant radius, which is represented by the equation r = constant. In this case, the constant is 16. Therefore, statement (iii) is false.

Based on our analysis, the correct answer is A. (i) and (iii) are the true statements.

2) To determine the number of petals in the combined rose curves, we need to find the values of θ that generate petals in both equations.

For r = 9 cos 61θ, the coefficient of θ is 61. Dividing this by 2 gives us 30.5, which means there are 30 petals for this equation.

For r = 8 sin 60θ, the coefficient of θ is 60. Dividing this by 2 gives us 30, which means there are 30 petals for this equation.

To find the total number of petals, we add the number of petals from each equation: 30 + 30 = 60 petals.

Therefore, the correct answer is not listed. The combined rose curves have 60 petals.

18) To analyze the effect of replacing r with -r and θ with π - θ, let's consider how these replacements affect the shape of the curve.

Replacing r with -r reflects the curve across the origin. This means that if a point (r, θ) lies on the curve, the point (-r, θ) will also lie on the curve. This transformation does not alter the shape or symmetry of the curve.

Replacing θ with π - θ reflects the curve across the y-axis. This means that if a point (r, θ) lies on the curve, the point (r, π - θ) will also lie on the curve. This transformation alters the shape and symmetry of the curve.

Therefore, if the shape of the curve is unaffected when r is replaced by -r and θ is replaced by π - θ, the curve must be symmetric with respect to the y-axis.

Hence, the correct answer is D. It's symmetric with respect to the y-axis.