Question

Given cosθ=3/4 and angleθ is in Quadrant IV, what is the exact value of sinθ in simplest form? Simplify all radicals if needed.

Answer 1

To determine the exact value of sinθ given cosθ=3/4 in Quadrant IV, we use the Pythagorean identity. Since sine is negative in the fourth quadrant, the exact value of sinθ is -√7/4.

Step-by-step explanation:

To find the exact value of sinθ when given cosθ=3/4 and angle θ is in Quadrant IV, we use the Pythagorean identity which states that sin²θ + cos²θ = 1. Since we know cosθ, we can rearrange this to find sinθ as follows:

1. sin²θ = 1 - cos²θ
2. sin²θ = 1 - (3/4)²
3. sin²θ = 1 - 9/16
4. sin²θ = 16/16 - 9/16
5. sin²θ = 7/16
6. sinθ = ±√(7/16)
7. sinθ = ±√7 / 4

In Quadrant IV, the sine function is negative, so we choose the negative root:

sinθ = -√7 / 4

Therefore, the exact value of sinθ in its simplest form is -√7 / 4.