What if you were given a right triangle and told that its sides measure 3, 4, and 5 inches? How could you find the sine, cosine, and tangent of one of the triangle's non-right angles? After completing this Concept, you'll be able to solve for these trigonometric ratios.

### Watch This

CK-12 Foundation: The Trigonometric Ratios

Watch the parts of the video dealing with the sine, cosine, and tangent.

James Sousa: Introduction to Trigonometric Functions Using Triangles

### Guidance

**Trigonometry** is the study of the relationships between the sides and angles of right triangles. The legs are called ** adjacent** or

**depending on which**

*opposite***angle is being used.**

*acute*

The three basic trigonometric ratios are called sine, cosine and tangent. For right triangle , we have:

**Sine Ratio:** or

**Cosine Ratio:** or

**Tangent Ratio:** or

An easy way to remember ratios is to use SOH-CAH-TOA.

**A few important points:**

- Always
**reduce ratios**(fractions) when you can. - Use the Pythagorean Theorem to find the missing side (if there is one).
- If there is a radical in the denominator,
**rationalize the denominator.**

#### Example A

Find the sine, cosine and tangent ratios of .

First, we need to use the Pythagorean Theorem to find the length of the hypotenuse.

#### Example B

Find the sine, cosine, and tangent of .

Find the length of the missing side.

#### Example C

Find the sine, cosine and tangent of .

This is a 30-60-90 triangle. The short leg is 6, and .

CK-12 Foundation: The Trigonometric Ratios

### Guided Practice

Answer the questions about the following image. Reduce all fractions.

1. What is ?

2. What is ?

3. What is ?

**Answers:**

1.

2.

3.

### Practice

Use the diagram to fill in the blanks below.

From questions 1-6, we can conclude the following. Fill in the blanks.

- and .
- and are _________ of each other.

Find the sine, cosine and tangent of . Reduce all fractions and radicals.