# Pythagorean Identities Worksheets

What are the Pythagorean Identities? A trigonometric identity refers to an equation with trigonometric functions, and that stands true for every value substituted for a variable. Trigonometric identities help in simplifying trigonometric expressions. Trigonometric identities involving the Pythagorean theorem are the most commonly used ones. In the unit circle, i.e., the circle with a radius of 1, a point on a unit circle (vertex of a right triangle) can be represented by cos(θ) and sin (θ). Now, the adjacent and opposite of right triangle has values of sin(θ) and cos(θ), the Pythagorean theorem can be applied to obtain . Sin 2(θ) + cos 2(θ) = 1. This equation is known as first Pythagorean identity. It stands true for all values of theta in a unit circle. By using the first Pythagorean identity, we can obtain other identities. Sin 2(θ) + cos 2(θ) = 1. Dividing each term by cos 2(θ). Sin2(θ)/cos2(θ) + cos2(θ) / cos2(θ) = 1 / cos2(θ). We know 1/cos(θ)=sec(θ) and sin(θ)/cos(θ)= tan(θ). Simplifying we term, we get: Tan2(θ) + 1= sec2(θ). We now have our second Pythagorean identity. Tan2(θ) + 1 =sec2(θ). Using the first identity to obtain the third Pythagorean identity. Sin2(θ) + cos2(θ) = 1. Dividing each term by sin2(). Sin2(θ)/sin2(θ) + cos2(θ)/sin2(θ) = 1/sin2(θ). We know that 1/sin(θ) =cosec(θ) and cos(θ)/sin(θ) =cot(θ). 1 + cot2(θ) = cosec2(θ). The third Pythagorean identity is: 1 + cot2(θ) = cosec2(θ).

• ### Basic Lesson

Guides students through solving equations that involve Pythagorean Identities. Demonstrates answer checking.

• ### Intermediate Lesson

Demonstrates how to solve more difficult problems.

• ### Independent Practice 1

A really great activity for allowing students to understand the concept of Pythagorean Identities.

• ### Independent Practice 2

Students find the Pythagorean Identities in assorted problems. The answers can be found below.

• ### Homework Worksheet

Students are provided with problems to achieve the concepts of Pythagorean Identities.

• ### Skill Quiz

This tests the students ability to evaluate Pythagorean Identities.

• ### Answer Key

Answers for math worksheets, quiz, homework, and lessons.

#### Outgoing Math

Q: How can you tell that a mathematician is extroverted?

A: When talking to you, he looks at your shoes instead of at his.