Finding the values for trig functions is pretty familiar to you by now. The trig functions of some particular angles may even seem obvious, since you've worked with them so many times. In some cases, you might be able to use this knowledge to your benefit to make calculating the values of some trig equations easier. For example, if someone asked you to evaluate

without consulting a table of trig values, could you do it?

You might notice right away that this is equal to four times . Can this help you? Read this Concept, and at its conclusion you'll know how to use certain formulas to simplify multiples of familiar angles to solve problems.

### Watch This

James Sousa: Double Angle Identities

### Guidance

Here we'll start with the sum and difference formulas for sine, cosine, and tangent. We can use these identities to help derive a new formula for when we are given a trig function that has twice a given angle as the argument. For example, . This way, if we are given and are asked to find , we can use our new double angle identity to help simplify the problem. Let's start with the derivation of the double angle identities.

One of the formulas for calculating the sum of two angles is:

If and are both the same angle in the above formula, then

This is the double angle formula for the sine function. The same procedure can be used in the sum formula for cosine, start with the sum angle formula:

If and are both the same angle in the above formula, then

This is one of the double angle formulas for the cosine function. Two more formulas can be derived by using the Pythagorean Identity, .

and likewise

Therefore, the double angle formulas for are:

Finally, we can calculate the double angle formula for tangent, using the tangent sum formula:

If and are both the same angle in the above formula, then

We can use these formulas to help simplify calculations of trig functions of certain arguments.

#### Example A

If and is in Quadrant II, find , , and .

**
Solution:
**
To use
, the value of
must be found first.

.

However since is in Quadrant II, is negative or .

For , use

For , use . From above, .

#### Example B

Find .

**
Solution:
**
Think of
as
.

Now, use the double angle formulas for both sine and cosine. For cosine, you can pick which formula you would like to use. In general, because we are proving a cosine identity, stay with cosine.

#### Example C

Solve the trigonometric equation such that

**
Solution:
**
Using the sine double angle formula:

### Vocabulary

**
Double Angle Identity:
**
A
**
double angle identity
**
relates the a trigonometric function of two times an argument to a set of trigonometric functions, each containing the original argument.

### Guided Practice

1. If and is in Quad II, find the exact values of and

2. Find the exact value of

3. Verify the identity:

**
Solutions:
**

1. If and in Quadrant II, then cosine and tangent are negative. Also, by the Pythagorean Theorem, the third side is . So, and . Using this, we can find , and .

2. This is one of the forms for .

3. Step 1: Use the cosine sum formula

Step 2: Use double angle formulas for and

Step 3: Distribute and simplify.

### Concept Problem Solution

Since the problem wants you to find:

You can simplify this into a familiar angle:

And then apply the double angle identity:

### Practice

Simplify each expression so that it is in terms of and .

Solve each equation on the interval .

Simplify each expression so that only one calculation would be needed in order to evaluate.