You want to find the exact value of . How could you find this value without using a calculator?

### Guidance

In the previous concept, we added two different angles together to find the exact values of trig functions. In this concept, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles. Here we will introduce the Double-Angle and Half-Angle Formulas.

#### Double-Angle and Half-Angle Formulas

The signs of and depend on which quadrant lies in. For and any formula can be used to solve for the exact value.

#### Example A

Find the exact value of .

**
Solution:
**
is half of
and in the first quadrant.

#### Example B

Find the exact value of if and .

**
Solution:
**
To use the sine double-angle formula, we also need to find
, which would be
because
is in the
quadrant.

#### Example C

Find the exact value of for from Example B.

**
Solution:
**
Use
to solve for
.

**
Concept Problem Revisit
**

so we can use the formula for

If we simplify this expression, we get .

### Guided Practice

1. Find the exact value of .

2. and . Find:

a)

b)

**
Answers
**

1. is in the quadrant.

2. First, find . , so

a)

b) You can use either formula.

### Vocabulary

- Double-Angle and Half-Angle Formulas

### Practice

Find the exact value of the following angles.

The and . Find:

The and . Find: