# Using the Unit Circle with Trigonometric Identities

Using the Unit Circle with Trigonometric Identities - Trigonometry can sometimes be difficult to grasp. Trigonometric terms like sine, cosine, and tangent can seem like abstract terms and be challenging to grasp conceptually. The unit circle helps significantly in understanding trigonometry and the use of the functions Definition Of Cosine, Sine, And Tan Sine and cosine are generally defined as: Sin(θ)= opposite/ hypotenuse, Cos(θ)=adjacent/hypotenuse, Tan(θ)= sin(θ)/cos(θ) Adjacent is defined as the length of the side next to the right angle, the opposite is the length of the side at the opposite of the angle, and the hypotenuse is defined as the longest side of the diagonal of the triangle. Suppose a triangle with hypotenuse as the radius of the unit circle; this means that with hypotenuse =1, the above equations will become Sin(θ) = opposite/1 = opposite, Cos(θ) =adjacent/1 = adjacent. If we consider the angle at the center of the circle, then the y-axis becomes the opposite, and the x-axis becomes the adjacent side. In simpler terms, cosine returns the x-coordinate for the given angle and sine returns the y-coordinate. That means, sin(0) = 0 and cos(0) = 1. Similarly, sin(90) = 1 and cos(90) = 0, as this is the point with coordinates y=1 and x=0. In equation form: Cos(θ)= x, Sin(θ) = y. Tan is defined as: Tan(θ) = sin(θ) / cos(θ). But the unit circle definition of sine and cosine, it is defined as: Tan(θ) = opposite/ adjacent. In terms of coordinates: Tan(θ) = y/x. These worksheets and lessons show students how to practically apply the use of the Unit Circle.

### Aligned Standard: HSF-TF.A.2

- Hello Unit Circle Step-by-step Lesson - Why not start with the very simple and work up?
- Guided Lesson - Are we completely focused on the hypotenuse? Yes, we are!
- Guided Lesson Explanation - We focus again on the simple use of the Unit Circle.
- Practice Worksheet - To progress to the next level, check out the member materials.
- Matching Worksheet - These might seem simple, but it is all about Unit Circle application.

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

### Homework Sheets

You are asked to find the values of fixed trig. ratios.

- Homework 1 - Example Let cos Θ =- 3/5
- Homework 2 - Find the value of a given trigonometric ratio using unit circles: cos Θ =, tan Θ =, sec Θ =, csc Θ = .
- Homework 3 - If the sin Θ = 2/5 the trigonometric is what?

### Practice Worksheets

Having a large unit circle handy is always a plus with this topic.

- Practice 1 - There are four problems for each question.
- Practice 2 - Learn how to use trigonometric ratios.
- Practice 3 - Use the unit circle as a reference tool.

### Math Skill Quizzes

Find all the trig. values for Θ.

- Quiz 1 - Use the reference tool to help you along.
- Quiz 2 - These are simple and easier forms of these problems.
- Quiz 3 - The more advanced of the quizzes.