# 1.23: Quotient Identities

**At Grade**Created by: CK-12

**Practice**Quotient Identities

You are working in math class one day when your friend leans over and asks you what you got for the sine and cosine of a particular angle.

"I got

"It looks like I'm supposed to calculate the tangent function for the same angle you just did, but I can't remember the relationship for tangent. What should I do?" he says.

Do you know how you can help your friend find the answer, even if both you and he don't remember the relationship for tangent?

Keep reading, and by the end of this Concept, you'll be able to help your friend.

### Watch This

The middle portion of this video reviews the Quotient Identities.

James Sousa: The Reciprocal, Quotient, and Pythagorean Identities

### Guidance

The definitions of the trig functions led us to the reciprocal identities, which can be seen in the Concept about that topic. They also lead us to another set of identities, the quotient identities.

Consider first the sine, cosine, and tangent functions. For angles of rotation (not necessarily in the unit circle) these functions are defined as follows:

Given these definitions, we can show that

The equation

#### Example A

If

**Solution:**

#### Example B

Show that

**Solution:**

#### Example C

If

**Solution:**

### Guided Practice

1. If

2. If

3. If

**Solutions:**

1.

2.

3.

### Concept Problem Solution

Since you now know that:

you can use this knowledge to help your friend with the sine and cosine values you measured for yourself earlier:

### Explore More

Fill in each blank with a trigonometric function.

tanθ=sinθ? cosθ=sinθ? cotθ=?sinθ cosθ=(cotθ)⋅(?) - If
cosθ=513 andsinθ=113 , what is the value oftanθ ? - If
sinθ=35 andcosθ=45 , what is the value oftanθ ? - If
cosθ=725 andsinθ=2425 , what is the value oftanθ ? - If
sinθ=1237 andcosθ=3537 , what is the value oftanθ ? - If
cosθ=2029 andsinθ=2129 , what is the value oftanθ ? - If
sinθ=3989 andcosθ=8089 , what is the value oftanθ ? - If
cosθ=4873 andsinθ=5573 , what is the value oftanθ ? - If
sinθ=6597 andcosθ=7297 , what is the value oftanθ ? - If
cosθ=12 andcotθ=3√3 , what is the value ofsinθ ? - If
tanθ=0 andcosθ=−1 , what is the value ofsinθ ? - If
cotθ=−1 andsinθ=−2√2 , what is the value ofcosθ ?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.23.

### Image Attributions

## Description

## Learning Objectives

Here you'll learn what a quotient identity is and how to derive it.

## Difficulty Level:

At Grade## Authors:

## Subjects:

## Search Keywords:

## Concept Nodes:

## Date Created:

Sep 26, 2012## Last Modified:

Feb 26, 2015## Vocabulary

**Save or share your relevant files like activites, homework and worksheet.**

To add resources, you must be the owner of the Modality. Click Customize to make your own copy.