You are given the following information about

What are and ?

### Trigonometric Identities

You can use the Pythagorean, Tangent and Reciprocal Identities to find all six trigonometric values for certain angles. Let’s walk through a few examples so that you understand how to do this.

#### Solve the following problems using trigonometric identities

Given that and , find .

Use the Pythagorean Identity to find .

Because is in the first quadrant, we know that sine will be positive.

Find of from Example A.

Use the Tangent Identity to find .

Find the other three trigonometric functions of from Example.

To find secant, cosecant, and cotangent use the Reciprocal Identities.

### Examples

#### Example 1

Earlier, you were asked what are and of .

First, use the Pythagorean Identity to find .

However, because is restricted to the second quadrant, the cosine must be negative. Therefore, .

Now use the Tangent Identity to find .

Find the values of the other five trigonometric functions.

#### Example 2

First, we know that is in the second quadrant, making sine positive and cosine negative. For this problem, we will use the Pythagorean Identity to find secant.

If , then . because the numerator value of tangent is the sine and it has the same denominator value as cosine. and from the Reciprocal Identities.

#### Example 3

is in the third quadrant, so both sine and cosine are negative. The reciprocal of , will give us . Now, use the Pythagorean Identity to find cosine.

and

### Review

- In which quadrants is the sine value positive? Negative?
- In which quadrants is the cosine value positive? Negative?
- In which quadrants is the tangent value positive? Negative?

Find the values of the other five trigonometric functions of .

- Aside from using the identities, how else can you find the values of the other five trigonometric functions?
- Given that and is in the quadrant, what is ?
- Given that and is in the quadrant, what is ?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.7.