Trigonometry Help

Question

Using a right triangle diagram, rewrite cos(sin^-1(1+x)) as an algebraic function of x.

Answer 1

Let θ = sin^-1(1+x).  So, sinθ = 1+x = (1+x) / 1.

 

Draw a right triangle with hypotenuse 1, and side opposite acute angle θ of length 1+x.

 

Then, by the Pythagorean Theorem, the side adjacent to θ has length √[1² - (1+x)²] = √(-2x-x²).

 

So, cos(sin^-1(1+x)) = cosθ = adjacent/hypotenuse = √(-2x-x²)