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Trigonometry and the Pythagorean Theorem

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  • 0:11 SohCahToa and…
  • 0:58 Inverse and Reciprocation
  • 1:47 Using Trigonometry and…
  • 4:07 Lesson Summary
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Lesson Transcript
Instructor: Eric Garneau
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.

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
pythagorean theorem

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
Trig Triangle Example 1

Using Trigonometry and Geometry

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.

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