Question

I don’t know where to start with this problem

Answer 1

√(4/5)

Explanation:

First, let's use reflection property to find tan θ.

tan(-θ) = 1/2

-tan θ = 1/2

tan θ = -1/2

Since tan θ < 0 and sec θ > 0, θ must be in the fourth quadrant.

Now let's look at the problem we need to solve:

sin(5π/2 + θ)

Use angle sum formula:

sin(5π/2) cos θ + sin θ cos(5π/2)

Sine and cosine have periods of 2π, so:

sin(π/2) cos θ + sin θ cos(π/2)

Evaluate:

(1) cos θ + sin θ (0)

cos θ

We need to write this in terms of tan θ. We can use Pythagorean identity:

1 + tan² θ = sec² θ

1 + tan² θ = (1 / cos θ)²

±√(1 + tan² θ) = 1 / cos θ

cos θ = ±1 / √(1 + tan² θ)

Plugging in:

cos θ = ±1 / √(1 + (-1/2)²)

cos θ = ±1 / √(1 + 1/4)

cos θ = ±1 / √(5/4)

cos θ = ±√(4/5)

Since θ is in the fourth quadrant, cos θ > 0. So:

cos θ = √(4/5)

Or, written in proper form:

cos θ = (2√5) / 5