Algebraic Expressions

Question

write an algebraic expression for y-tan (cos^-1 x/3) for 0<x<3.

Answer 1

Let θ = cos^-1(x/3).  Then cosθ = x/3

 

   Draw a right triangle with acute angle θ, side adjacent to θ labelled as    x, and hypotenuse labelled as 3 (so cosθ = x/3).  Then, by the              Pythagorean Theorem, the side opposite θ is √(9-x²).  

 

Therefore, tan(cos^-1(x/3)) = tanθ = opposite/adjacent = √(9-x²)/x