Like?

Trigonometry and the Pythagorean Theorem

Start Your Free Trial To Continue Watching
As a member, you'll also get unlimited access to over 8,500 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.
Free 5-day trial
It only takes a minute. You can cancel at any time.
Already registered? Login here for access.
Start your free trial to take this quiz
As a premium member, you can take this quiz and also access over 8,500 fun and engaging lessons in math, English, science, history, and more. Get access today with a FREE trial!
Free 5-day trial
It only takes a minute to get started. You can cancel at any time.
Already registered? Login here for access.
  1. 0:11 SohCahToa and Pythagorean Theorem
  2. 0:58 Inverse and Reciprocation
  3. 1:47 Using Trigonometry and Geometry
  4. 4:07 Lesson Summary
Show Timeline
Taught by

Erin Lennon

Erin has taught math and science from grade school up to the post-graduate level. She holds a Ph.D. in Chemical Engineering.

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.

Unlock Content Over 8,500 lessons in all major subjects

Get FREE access for 5 days,
just create an account.

Start a FREE trial

No obligation, cancel anytime.

Want to learn more?

Select a subject to preview related courses:

People are saying…

"This just saved me about $2,000 and 1 year of my life." — Student

"I learned in 20 minutes what it took 3 months to learn in class." — Student

See more testimonials

Did you like this?
Yes No

Thanks for your feedback!

What didn't you like?

What didn't you like?

Congratulations! You've reached the last video in the chapter.
Start the Next Chapter
Create your Account

Sign up now for your account. Get unlimited access to 8,500 lessons in math, English, science, history, and more.

Meet Our Instructors

Meet all 53 of our instructors

Copyright