Explore how the Pythagorean Theorem can be used in conjunction with trigonometric functions. In this lesson, take an inverse trigonometric function, and define all three sides of a right triangle.
SohCahToa and the Pythagorean Theorem
The Pythagorean Theorem applies to right triangles
Let's take one more look at trigonometry. Remember there are two things that you need to keep in mind with respect to trigonometry. You need to remember SohCahToa - sin(theta) equals the opposite over the hypotenuse, cos(theta) equals the adjacent over the hypotenuse, and tan(theta) equals the opposite over the adjacent. So here's our right triangle. We've got theta, the adjacent leg, the opposite leg and the hypotenuse.
The second thing you need to remember is the Pythagorean Theorem. That says that if you have this abc right triangle, a^2 + b^2 = c^2. With those two things, you can do a lot of important calculations in calculus and geometry.
Inverse and Reciprocation
What are some of the things you might need to know? Well, 1/sin(theta) is known as csc(theta). It's equal to the hypotenuse over the opposite side. We will never write this as sin^-1(theta). Why is that? Well, sin^-1(theta) is really the inverse function of sin(theta); it's not 1/sin(theta). So what this means is that sin^-1(sin(theta)) will give you theta, just like f^-1(f(x)) will give back x. That's the definition of the inverse function here. This also means that if sin(theta) is the opposite over the hypotenuse, the cosecant sin^-1 of the opposite over the hypotenuse is equal to theta.
Some of the ways to represent this right triangle
Alright, so what's an example of using the Pythagorean Theorem, sines and cosines in a meaningful way? Let's say you have the function sin(y) = x. This might come from having the equation csc(x) = y. That's like saying sin^-1(x) = y. So sin(y) = x. Sin(y) also equals the opposite divided by the hypotenuse. So let's draw out a right triangle and let's make our angle y. Here I've got the opposite side and here I've got the hypotenuse. I know that the opposite divided by the hypotenuse is equal to x, so why don't I just call the hypotenuse 1 and this opposite side equal to x? Opposite over hypotenuse is equal to x divided by 1, so this triangle makes sense with our equation sin(y) = x.
From the Pythagorean Theorem we can find out what this other side equals. I know that x^2 plus the length of this side squared has to equal 1. That means that this side is equal to the square root of 1 - x^2, so let's put that in our triangle. Now this triangle represents sin(y) = x. Now that we know that, we can find out what the cosine and tangent of y are. The cosine of y is equal to the adjacent side divided by the hypotenuse. The adjacent side is the square root of 1 - x^2, and the hypotenuse is just 1, so here's my cosine of y. Similarly, the tangent of y equals the opposite over the adjacent, which is just x divided by the square root of 1 - x^2. All of these represent this right triangle.
Triangle for the final example
You can do the same thing if you have cos(y) = x. This might have come from sec(x) = y, so if cos(y) = x that's like saying x is equal to the adjacent over the hypotenuse. So let's make the hypotenuse 1 again, and the adjacent side equal to x. The opposite side, by the Pythagorean Theorem, equals the square root of 1 - x^2, and I can find sine of y and tangent of y based solely on this.
This is going to become very useful if you can remember it throughout all of calculus. It's really just two rules: 1) SohCahToa and 2) the Pythagorean Theorem. Remember these and you're set.