Question

Trig Identities---

1. Use the pythagorean identity
Find cotθ if cosθ = 1/7 and the value exists in the first quadrant.

I have a calculus exam tomorrow and have no idea how to solve this equation, please provide a detailed explanation. I'm slow.

Answer 1

Let's first recall the Pythagorean identity:

sin²θ + cos²θ = 1

We can use this identity to find the value of sinθ, given that cosθ = 1/7 and θ is in the first quadrant. Since θ is in the first quadrant, both sinθ and cosθ are positive.

cosθ = 1/7

cos²θ = (1/7)² = 1/49

sin²θ + cos²θ = 1

sin²θ + 1/49 = 1

sin²θ = 1 - 1/49 = 48/49

sinθ = √(48/49) = (4/7)√3

Now we can use the definition of cotangent to find cotθ:

cotθ = cosθ/sinθ

Substituting the values we found for cosθ and sinθ, we get:

cotθ = (1/7)/[(4/7)√3] = √3/4

Therefore, cotθ = √3/4 when cosθ = 1/7 and θ is in the first quadrant.