# Precalculus : Trigonometric Identities

## Example Questions

### Example Question #1 : Sum And Difference Identities For Sine

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:

### Example Question #1 : Sum And Difference Identities For Sine

Evaluate

.

Explanation:

The angle  or .

Using the first one:

We can find these values in the unit circle:

### Example Question #1 : Sum And Difference Identities For Cosine

Evaluate the exact value of:

Explanation:

In order to solve , two special angles will need to be used to solve for the exact values.

The angles chosen are  and  degrees, since:

Write the formula for the cosine additive identity.

Substitute the known variables.

### Example Question #2 : Sum And Difference Identities For Cosine

Find the exact value of .

Explanation:

To solve , we will need to use both the sum and difference identities for cosine.

Write the formula for these identities.

To solve for  and , find two special angles whose difference  and sum equals to the angle 15 and 75, respectively.  The two special angles are 45 and 30.

Substitute the special angles in the formula.

Evaluate both conditions.

Solve for .

### Example Question #3 : Sum And Difference Identities For Cosine

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 #4 : Sum And Difference Identities For Cosine

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 #1 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve the following equation for :

Explanation:

Use the trigonometric identities to switch sec into terms of tan:

hence,

So we have , making

Therefore the solution is  for n being any integer.

### Example Question #2 : Solving Trigonometric Equations And Inequalities

Which of the following is not a solution to   for

Explanation:

We begin by setting the right side of the equation equal to 0.

The equation might be easier to factor using the following substitution.

This gives the following

This can be factored as follows

Therefore

Replacing our substitution therefore gives

Within our designated domain, we get three answers between our two equations.

when

when

Therefore, the only choice that isn't correct is

### Example Question #1 : Solving Trigonometric Equations And Inequalities

Find one possible value of .

Explanation:

Begin by isolating the tangent side of the equation:

Next, take the inverse tangent of both sides:

Divide by five to get the final answer:

### Example Question #1 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve for the angle value.