Visualizing Pythagorean Identities (Updated)

Author:
HaileyMF
You’ve seen this image before, but with only a single triangle – the purple triangle. You know on the unit circle, the x-coordinate corresponds with the cosine of the angle, and the y-coordinate corresponds with the sine of the angle. From that simple purple triangle, you can see that we have (using the Pythagorean Theorem): But you’ve never seen that BLUE triangle or that RED triangle. It turns out the lengths of the legs of the other triangles actually are those other trigonometric functions. You’re going to discover that, and start finding some neat trigonometric identities (like that Pythagorean Identity) visually.
1.) Why can I conclude that all four triangles are similar? What is the mathematical justification for that conclusion? 2.) Find the side lengths for all of the triangles in terms of simple trigonometric functions and/or "1". 3.) Use the Pythagorean Theorem on all five triangles to come up with at least four equations. You don’t need to simplify/expand.