You've gotten quite good at knowing the values of trig functions. So much so that you and your friends play a game before class everyday to see who can get the most trig functions of different angles correct. However, your friend Jane keeps getting the trig functions of more angles right. You're amazed by her memory, until she smiles one day and tells you that she's been fooling you all this time.

"What you do you mean?" you say.

"I have a trick that lets me calculate more functions in my mind by breaking them down into sums of angles." she replies.

You're really surprised by this. And all this time you thought she just had an amazing memory!

"Here, let me show you," she says. She takes a piece of paper out and writes down:

"This looks like an unusual value to remember for a trig function. So I have a special rule that helps me to evaluate it by breaking it into a sum of different numbers."

By the end of this Concept, you'll be able to calculate the above function using a special rule, just like Jane does.

### Watch This

James Sousa: Sum and Difference Identities for Sine

### Guidance

Our goal here is to figure out a formula that lets you break down a the sine of a sum of two angles (or a difference of two angles) into a simpler formula that lets you use the sine of only one argument in each term.

To find :

In conclusion,
, which is the
*
sum
*
formula for sine.

To obtain the identity for :

In conclusion,
, so, this is the
*
difference
*
formula for sine.

#### Example A

Find the exact value of

**
Solution:
**
Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.

#### Example B

Given , where is in Quadrant II, and , where is in Quadrant I, find the exact value of .

**
Solution:
**
To find the exact value of
, here we use
. The values of
and
are known, however the values of
and
need to be found.

Use , to find the values of each of the missing cosine values.

For , substituting transforms to or , however, since is in Quadrant II, the cosine is negative, .

For use and substitute or and and since is in Quadrant I,

Now the sum formula for the sine of two angles can be found:

#### Example C

Find the exact value of

**
Solution:
**
Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.

### Vocabulary

**
Sine Sum Formula:
**
The
**
sine sum formula
**
relates the sine of a sum of two arguments to a set of sine and cosines functions, each containing one argument.

**
Sine Difference Formula:
**
The
**
sine difference formula
**
relates the sine of a difference of two arguments to a set of sine and cosines functions, each containing one argument.

### Guided Practice

1. Find the exact value for

2. Find the exact value for

3. If , is in quad III, and , is in quad II find

**
Solutions:
**

1.

2.

3.

If and in Quadrant III, then cosine is also negative. By the Pythagorean Theorem, the second leg is , so . If the and in Quadrant II, then the cosine is also negative. By the Pythagorean Theorem, the second leg is , so . To find , plug this information into the sine sum formula.

### Concept Problem Solution

With the sine sum formula, you can break the sine into easier to calculate quantities:

### Practice

Find the exact value for each sine expression.

Write each expression as the sine of an angle.

- Prove that
- Suppose that x, y, and z are the three angles of a triangle. Prove that
- Prove that
- Prove that
- Prove that