Find the maximum value of |r| and any zeros of r. Assume 0<=(theta)<2pi

Question

 r = 8 sin (2θ) 

 r = 9 + 18 cos θ

r = 8 cos 3θ

Answer 1

Since the range of cosine or sine is ±1, the following maximums are:

r =|8 sin (2θ)| max = 8

r = |9 + 18 cos θ| max = 9+18 = 27

r = |8 cos 3θ| max = 8

Zeros: When r = 0

1) when 2θ = kπ, k ≥ 0 --> θ = kπ/2 = 0, π/2, π, 3π/2

2) when cos(θ) = -1/2 --> θ = 2π/3, 4π/3

3) when 3θ = kπ/2 , k ≥1 --> θ=kπ/6 = π/6, π/3, π/2, 2π/3, 5π/3

Answer 2

Max[sin]=max[cos]=1

And t=angle theta

1) max=8

2t=0 or 2t=pi

T=0 or t=pi/2

2) max=27

Cos t = -1/2

T= 120 or t=240

Or t= (2/3)pi or t= (4/3)pi

3) max=8

3t= pi/2 or 3t=(3/2)pi

T= pi/6 or t= pi/2