Calculus Question?

Question

You are given that tan(y)=x

Find sin(y)^2

Answer 1

tan(y) = x

Draw a triangle. Make "y" the angle. Since "x" is the tangent of angle "y," and tangent (which is opposite/adjacent) equals "x/1." Now, using Pythagorean's Theorem, find the sine of angle "y:"

sin(y) = x/(sqrt(1 + x²))

sin²(y) = (x²) / (1 + x²)

Answer 2

If tan y = x, then the opposite leg = x and the adjacent leg = 1. Hence, the hypoteneuse is sqrt(x^2 + 1).

So (sin y)^2 = [x/sqrt(x^2 + 1)]^2 = x^2/(x^2 + 1)

Answer 3

sin(y) / cos(y) = x, so

sin(y) = x cos(y).

To compute cos(y) in terms of x, use SOHCAHTOA. tan(y) = opposite / adjacent, so consider a right triangle with opposite = x and adjacent = 1.

hypotenuse = sqrt(1+x^2), so cos(y) = adjacent / hypotenuse = 1/sqrt(1+x^2).

Therefore sin(y)^2 = x^2 / (1+x^2).

Answer 4

tan^2y = (sin^2y)/cos^2y) = (sin^2y)/(1-sin^2y) = x^2
(sin^2y) / (1-sin^2y) = x^2
(sin^2y) = (x^2)(1-sin^2y)
sin^2y = (x^2-(x^2)sin^2y
sin^2y(1+x^2) = x^2
sin^2y = (x^2)/(x^2 + 1)

sin(y^2) = sin((arcsin(sqrt(x^2/(x^2 + 1)))^2)