Final answer:
To find cosθ with cscθ being 5/2, we use the Pythagorean identity. With sinθ as 2/5, cosθ is found to be ±√21/5 as the exact values.
Step-by-step explanation:
To find the exact values of cosθ if cscθ=5/2, we can use trigonometric identities and relationships. The cosecant function is the reciprocal of the sine function, so cscθ=5/2 implies that sinθ=2/5. To find cosθ, we can use the Pythagorean identity sin²θ + cos²θ = 1. Substituting sinθ=2/5 into the identity, we get (2/5)² + cos²θ = 1. Solving for cosθ yields:
- cos²θ = 1 - (2/5)²
- cos²θ = 1 - 4/25
- cos²θ = 21/25
- cosθ = ±√(21/25)
- cosθ = ±√21/5
Hence, the exact values of cosθ are ±√21/5.