Question

given cosθ = negative square root 21/5 and angle θ is in quadrant 3, what is the exact value of sin θ in simplest form? Simplify all radicals if needed.

Answer 1

The exact value of sin θ, given that cosθ = -√21/5 and angle θ is in quadrant 3, is -2/5.

Step-by-step explanation:

To find the exact value of sin θ when cosθ is given as negative square root 21/5 and angle θ is in quadrant 3, we can use the Pythagorean identity sin²θ + cos²θ = 1. Since we know cosθ, we can solve for sinθ.

In quadrant 3, both sine and cosine are negative, so we want the negative solution for sine. The cosθ given is -√21/5. Squaring this, we get cos²θ = 21/25. Substituting into the identity, we get sin²θ = 1 - cos²θ = 1 - 21/25 = 4/25.

Taking the square root gives us sinθ = ±√4/25, and since we're in quadrant 3 where sine is negative, sinθ = -√4/25, which simplifies to -2/5.