Trig Identities

Question

What is the cosine of (theta) if the sine of (theta) is equal to √3/2 and (theta) lies in the first quadrant?

Answer 1

With a quick sketch, this problem becomes super easy to solve!

1. Start by drawing a right triangle in the first quadrant with ∅ in standard position.
2. Since sine is opposite over hypotenuse, label the opposite side with √3 and the hypotenuse with 2.
3. If you're familiar with your special right triangles, you'll recognize this one as a 30-60-90, thus making the leg adjacent to theta equal to one and cos(∅) = 1/2
4. If you don't recognize the special right triangle right away, you can still solve this by doing the Pythagorean Theorem to find the missing side and apply that to get cos(∅)
5. Since ∅ is in the first quadrant, all trig functions are positive, leaving us with cos(∅) = 1/2

Answer 2

sin² θ + cos² θ = 1

Can you substitute value and solve for cosine?