Inverse Trig Triangle Method

Here’s something that becomes important for certain integrals in calc 2. Let’s simplify this expression fully, removing all trig:

The first thing we’re going to want to do is come up with a middleman variable (usually theta) to make this easier to visualize. Let’s set theta equal to the inverse tangent of x, and then substitute back in:

Now, let’s solve for x:

Here’s where the triangle comes into play. We know that the tangent of theta is x, so we’re going to invent a triangle where that’s true. I always start my triangles like this:

Time to remember SOH-CAH-TOA. The tangent is the opposite leg divided by the adjacent leg. We need those divided to equal x. The simplest way to do that is this:

We can find the third side using the Pythagorean theorem:

Now that we have all the sides, we can use SOH-CAH-TOA again to find the sine of theta:

This is possible with any combination of trig and inverse trig functions! Practice some more and check your work on the linked page :)