Write this as a function ASAP

Question

Write the √16-x² / x as a function of theta where 0<theta< π, by making the substitution x=4sintheta.

16-x² are square rooted and over x

Answer 1

I will solve this only because of its tricky nature. That is because the square root is always positive.

Thus, 16 - x^2 = 16 - (4 sin θ)^2 = 16 - 16 sin²θ = 16 cos²θ.

However, √(16cos²θ) = |4 cos θ|

On (0, 2π), sinθ > 0. Thus, the function will be defined throughout the interval

f(θ) = |4 cos θ|/4 sin θ = |cot θ|

(as sinθ > 0, we can write the absolute value for the fraction rather than just the numerator)