# Trigonometry : Identities with Angle Sums

## Example Questions

### Example Question #1 : Identities With Angle Sums

Given , what is ?

Explanation:

We need to use the formula

Substituting , and ,

### Example Question #2 : Identities With Angle Sums

Find the exact value of   using .

Explanation:

Our basic sum formula for cosine is:

Substituting the relevant angles gives us:

Now substitute in the exact values for each function, simplifying to keep radicals out of the denominator:

Multiply and subtract to obtain:

### Example Question #3 : Identities With Angle Sums

Find the exact value of the expression:

The expression is undefined.

Explanation:

There are two ways to solve this problem. If one recognizes the identity

,

the answer is as simple as:

If one misses the identity, or wishes to be more thorough, you can simplify:

### Example Question #4 : Identities With Angle Sums

Find the exact value of the expression:

Explanation:

The formula for the cosine of the difference of two angles is

Substituting, we find that

and

Therefore, what we are really looking for is

Thus,

### Example Question #5 : Identities With Angle Sums

Find the exact value of  using  and .

The quantity cannot be found exactly using the given information.

Explanation:

The sum identity for tangent states that

Substituting known values for  and , we have

For ease, multiply all terms by  to get .

At this point, multiply both halves of the fraction by the conjugate of the denominator:

Finally, simplify.

So, .

### Example Question #6 : Identities With Angle Sums

Suppose we have two angles,  and , such that:

Furthermore, suppose that angle  is located in the first quadrant and angle  is located in the fourth.

What is the measure of:

Explanation:

We can calculate some missing values using the pythagorean identities.

(Note the negative sign, because  is in the fourth quadrant, where the sine of the angle is always negative).

Note the positive value, since  is in the first quadrant, where cosine is positive.

Now using the rules for double angles:

And then the angle subtraction formula:

### Example Question #7 : Identities With Angle Sums

Calculate .

Explanation:

Recall the formula for the sine of the sum of two angles:

Here, we can evaluate  by noticing that  and applying the above formula to the sines and cosines of these two angles.

Hence,

### Example Question #8 : Identities With Angle Sums

What is the value of , using the sum formula.

Explanation:

The formula for

.

We can expand

,

where  and .

Substituting these values into the equation, we get

.

The final answer is -1, using what we know about the unit circle values.

### Example Question #9 : Identities With Angle Sums

Simplify the given expression.

Explanation:

This problem requires the use of two angle sum/difference identities:

Using these identities, we get

which simplifies to

which equals