Calculus writing in algebraic form

Question

Write the expression in algebraic form.

 

 

sin (cos^-1 x)

Answer 1

Picture a right triangle with an acute angle θ. Let that angle be the angle that has a cosine equal to x. Assign a value of x to the adjacent leg of that angle and the value 1 to the hypotenuse. That means cos(θ) = x or

θ=cos^-1(x). Using the Pythagorean Theorem, the side opposite θ is √(1-x²). That means the sin(θ) =

sin(cos^-1(x)) = √(1 - x²) / 1 = √(1- x²).

This Desmos graph shows the result working for values of x between -1 and 0, inclusive.

desmos.com/calculator/eaihpfb5un