Pythagorean Identities in Trigonometry: Definition & Examples

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  • 0:04 Pythagorean Identities…
  • 0:34 The First Identity
  • 1:40 The Second Identity
  • 2:17 The Third Identity
  • 2:55 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

There are three trigonometry identities based on of the Pythagorean theorem. You'll learn what they are in this lesson, as well as how to get from one to the other.

Pythagorean Identities Definition

The Pythagorean identities in trigonometry are the three identities that come from the Pythagorean theorem. Recall that the Pythagorean theorem states that the hypotenuse squared of a right triangle is the sum of the square of each of the other two sides, or a squared plus b squared equals c squared, like you can see here:

pythagorean identity

In the Pythagorean theorem, c stands for the hypotenuse, and a and b stand for the other two sides of the right triangle. From this theorem, three identities can be determined from substituting in sine and cosine.

The First Identity

The first one appearing here probably looks a lot like the Pythagorean theorem.

pythagorean identity

This identity comes from looking at the unit circle. All the right triangles formed by the unit circle will have a hypotenuse of 1. With a hypotenuse of 1 and with our angle at the origin of the coordinate plane on which the unit circle is drawn, we can determine some relationships between the sides and our sine and cosine.

The unit circle triangle with a hypotenuse of 1.
pythagorean identity

Looking at this triangle with a hypotenuse of 1, I see that my ratio for sin (theta) is a/1, so I know that a=sin (theta). My cosine then is b/1, so b=cos (theta), and my c=1. Plugging these values into the Pythagorean theorem, we arrive at our first identity.

This identity is useful when you see a problem with a sine squared plus a cosine squared. You can use this identity and replace the sine squared plus the cosine squared with the 1.

pythagorean identity

The example appearing here shows just one way you can use this identity. You can replace the sine squared plus cosine squared with the 1, or you can replace the 1 with sine squared plus cosine squared. It will depend on the problem and which will make the problem easier to solve.

The Second Identity

Our second identity actually comes from the first. This identity has one of the sides equaling 1.

pythagorean identity

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