Use a right triangle to write the following expression as an

Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x cos(sin^-1 5x) cos(sin^-1 5x) =

Solution

cos (sin^-15x)

Let sin^-15x=t

Taking sin on bith sides

sin(sin^-15x)=sin t

5x=sin t

sin t=5x=5x/1

cos (sin^-15x)=cos t

In right triangle,Sin theta= opposite/hypotenuse

We have sin t=5x/1

Therefore opposite=5x and hypotenuse=1

using pythagoras theorem

opposite²+adjacent²=hypotenuse²

(5x)²+ adjacent²=1^2

25x²+ adjacent²⁼1

Subtracting 25x² from both sides

25x²+ adjacent²=1-25x²

adjacent²⁼1-25x²

taking square root both sides

square root (adjacent²)=square root(1-25x²)

cos x= adjacent/hypotenuse

Therefore cos t= square root(1-25x²)/1

cos t=square root(1-25x²)

Substituting back sin^-15x for t

cos(sin^-15x)=square root(1-25x²)