# Precalculus : Sum and Difference Identities

## Example Questions

### Example Question #23 : Trigonometric Identities

In the problem below,  and .

Find

.

Explanation:

Since and is in quadrant I, we can say that and and therefore:

So .

Since  and  is in quadrant I, we can say that  and  and therefore:

.  So .

Using the cosine sum formula, we then see:

.

### Example Question #24 : Trigonometric Identities

In the problem below, and .

Find

.

Explanation:

Since and is in quadrant I, we can say that and and therefore:

So .

Since and is in quadrant I, we can say that and and therefore:

So .

Using the cosine difference formula, we see:

### Example Question #25 : Trigonometric Identities

Find  using the sum identity.

Explanation:

Using the sum formula for sine,

where,

yeilds:

.

### Example Question #26 : Trigonometric Identities

Calculate .

Explanation:

Notice that  is equivalent to . With this conversion, the sum formula can be applied using,

where

.

Therefore the result is as follows:

### Example Question #27 : Trigonometric Identities

Find the exact value for:

Explanation:

In order to solve this question, it is necessary to know the sine difference identity.

The values of  and must be a special angle, and their difference must be 15 degrees.

A possibility of their values that match the criteria are:

Substitute the values into the formula and solve.

Evaluate .

### Example Question #28 : Trigonometric Identities

Find the exact value of:

Explanation:

In order to find the exact value of , the sum identity of sine must be used.  Write the formula.

The only possibilites of  and  are 45 and 60 degrees interchangably. Substitute these values into the equation and evaluate.

### Example Question #29 : Trigonometric Identities

In the problem below, and .

Find

.

Explanation:

Since and is in quadrant I, we can say that and and therefore:

So .

Since and is in quadrant I, we can say that and and therefore:

So .

Using the sine sum formula, we see:

### Example Question #30 : Trigonometric Identities

In the problem below, and .

Find

.

Explanation:

Since and is in quadrant I, we can say that and and therefore:

So .

Since and is in quadrant I, we can say that and and therefore:

So .

Using the sine difference formula, we see:

### Example Question #11 : Sum And Difference Identities

Evaluate

.

Explanation:

is equivalent to or more simplified .

We can use the sum identity to evaluate this sine:

From the unit circle, we can determine these measures:

Evaluate

.