Trigonometry Polar Coordinates 5

Question 1

Find the area of the region inside r = 4 cos θ and outside r = 2.

Accepted Answer

4π/3 − √3

Answer

The intersections are at θ = ±π/3; integrating gives area = 4π/3 − √3.

Question 2

The polar curve r = θ (for θ ≥ 0) is called a(n):

Accepted Answer

Archimedean spiral

Answer

r = aθ defines an Archimedean spiral, where the distance between arms is constant.

Question 3

Convert r = 2/(3 − sin θ) to rectangular form and identify the conic.

Accepted Answer

Ellipse with e = 1/3

Answer

Dividing numerator and denominator by 3 gives r = (2/3)/(1 − (1/3)sin θ), so e = 1/3 < 1, an ellipse.

Question 4

At the pole, the curve r = sin θ cos θ has tangent lines. How many distinct tangent lines exist there?

Accepted Answer

2

Answer

r = 0 at θ = 0, π/2, π, 3π/2, but the tangent directions at θ = 0 and π are the same line, as are θ = π/2 and 3π/2, giving 2 distinct tangent lines.

Question 5

What is the Cartesian form of r = sec θ?

Accepted Answer

x = 1

Answer

r = sec θ means r = 1/cos θ → r cos θ = 1 → x = 1.

Trigonometry Practice Test