## Pythagorean Trigonometric Identities

#### Aligned To Common Core Standard:

**High School Trigonometric Functions** - HSF-TF.C.8

What Are Pythagorean Trigonometric Identities?
In mathematics, identity is referred to as an equation that stands true for all values
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)
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) These worksheets and lesson can help you solve and better understand the most common trigonometric identities.

### Printable Worksheets And Lessons

- 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.

#### 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.