Question

Find the sum of all values of θ such that θ∈[0,2π] and cos(θ)⋅4sec(θ)=8.

A. π/2

B. π

C. 3π/2

D. 2π

Answer 1

To find the sum of all values of θ such that cos(θ)·4sec(θ)=8, we simplify the equation and find the values of θ where cos(θ) = ½. In the given range, this occurs at θ = π/3 and θ = 5π/3, summing to 2π.

Step-by-step explanation:

The question asks for the sum of all values of θ such that θ∈[0,2π] and cos(θ)·4sec(θ)=8.

Firstly, let's simplify the equation.

cos(θ)·4sec(θ) = 4cos(θ)·4/cos(θ) = 16

This implies that cos(θ) must be equal to ½ since 16 × ½ = 8.

The values of θ that satisfy cos(θ) = ½ in the interval [0, 2π] are θ = π/3 and θ = 5π/3.

Summing these values, we get π/3 + 5π/3 = 2π.

Therefore, the correct answer is D. 2π.