Pre-cal Problem, please, help is very much appreciated!

Let y be the length of the third side. By the Pythagorean theorem,

y² + x² = 3² = 9

so that

y = √(9 - x²)

(taking the positive root because one expects length to be positive)

Then by the definitions of sine, cosine, and tangent, we have

sin(θ) = x / 3   →   θ = sin⁻¹(x / 3)

cos(θ) = √(9 - x²) / 3   →   θ = cos⁻¹(√(9 - x²) / 3)

tan(θ) = x / √(9 - x²)   →   θ = tan⁻¹(x / √(9 - x²))

yes

yes yes yes