# Precalculus : Trigonometric Identities

## Example Questions

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### Example Question #1 : Sum And Difference 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 tangent sum formula, we see:

### Example Question #2 : Sum And Difference 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 tangent sum formula, we see:

### Example Question #1 : Trigonometric Identities

Given that and , find .

Explanation:

Jump straight to the tangent sum formula:

From here plug in the given values and simplify.

### Example Question #1 : Trigonometric Identities

Which of the following expressions best represents ?

Explanation:

Write the identity for .

Set the value of the angle equal to .

Substitute the value of  into the identity.

### Example Question #5 : Sum And Difference Identities

Find  using the sum identity.

Explanation:

Using the sum formula for sine,

where,

yeilds:

.

### Example Question #6 : Sum And Difference 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 #1 : Sum And Difference 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 #1 : Sum And Difference 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 #9 : Sum And Difference 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 #10 : Sum And Difference 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:

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