# Pythagorean Trig Identities Worksheets

If you have a structure of any kind that you can certainly identify a right angle within it, you can learn a ton about the structure in terms of the length of sides and angles at which things are positioned within it through the use of trigonometric functions. Trigonometric functions that involve the use of the Pythagorean theorem are referred to as Pythagorean identities. These worksheets and lesson can help you solve and better understand the most common trigonometric identities.

### Aligned Standard: HSF-TF.C.8

- Missing Triangle Values Step-by-step Lesson - You are given two identities find all the rest of them.
- Guided Lesson - Simplifying complex expressions is quite a bit task for most people.
- Guided Lesson Explanation - Understanding the concept of inverses is three quarters of the battle here.
- Practice Worksheet - I made these a little bit more difficult than what you should see at the High School level.
- Matching Worksheet - Careful! Two of the answers are very close. Take your time with those.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

These problems are all over the concept brought up by the standard area.

- Homework 1 - You will need to match the known identities to expressions.
- Homework 2 - These can be worked through in a number of different ways.
- Homework 3 - Use the given triangle to help you solve the problem.

### Practice Worksheets

This section covers the majority of values that you will see on national exams.

- Practice 1 - Simplify the expression: Sec
^{2}x – cot x tan x to a single trigonometric function. - Practice 2 - If cos Θ = 8/14. Find the values of the cot Θ, using a Pythagorean identity.
- Practice 3 - Use a well known Pythagorean identity to solve this.

### Math Skill Quizzes

You really need to spend sometime reviewing the trig. Identities before tackling these.

- Quiz 1 - You should know be able to find these values quickly.
- Quiz 2 - What is the missing piece?
- Quiz 3 - How can you solve each of these with the fewest steps?

### What Are Pythagorean Trigonometric Identities?

In mathematics, identity is referred to as an equation that stands true for all values of the variables that reside within it. 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. If you have anything that has a right triangle within it, you can learn a great deal about it with the help of this form of math. These identities ca be used to understand the nature and relationships of all angles and sides within the structure.

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 such as:

Sin^{2}(θ) + cos^{2}(θ) = 1

Dividing each term by cos^{2}(θ)

Sin^{2}(θ)/cos^{2}(θ) +cos^{2}(θ)/cos^{2}(θ) = 1/cos^{2}(θ)

We know 1/cos(θ)=sec(θ) and sin(θ)/cos(θ)= tan(θ)

Simplifying we term, we get : Tan^{2}(θ) + 1= sec^{2}(θ)

We now have our second Pythagorean identity : Tan^{2}(θ) + 1 =sec^{2}(θ)

Using the first identity to obtain the third Pythagorean identity : Sin^{2}(θ) + cos^{2}(θ) = 1

Dividing each term by sin^{2}(θ)

Sin^{2}(θ)/sin^{2}(θ) + cos^{2}(θ)/sin^{2}(θ) = 1/sin^{2}(θ)

We know that 1/sin(θ) =cosec(θ) and cos(θ)/sin(θ) =cot(θ)

1 + cot^{2}(θ) = cosec^{2}(θ)

The third Pythagorean identity is : 1 + cot^{2}(θ) = cosec^{2}(θ).