trigonometry hw

Question

Use right triangles to write in algebraic form: cos(tan^-1 (y/8)). Assume that y is positive and that the given inverse trigonometric function is defined for the expression in y. Your answer should have no trigonometric functions.

Answer 1

First, let's focus on the first function which is inside the parenthesis: tan^-1(y/8)

If you think about a right triangle, the tan refers to the angle opposite/adjacent so we know the two sides of the right triangle: y and 8 which means the hypotenuse is √y²+8²)

tan^-1(y/8) represents one of the non-90 degree angle in the right triangle.

So cos(tan^-1(y/8)) = adjacent / hypotenuse = 8 / √y²+8²)

Answer 2

Draw and label a diagram!

tan^-1 (y / 8) ⇒ c = √(64 + y²)

cos tan^-1 (y / 8) = 8 / √(64 + y²)