# 3.7: Sine Sum and Difference Formulas

**At Grade**Created by: CK-12

**Practice**Sine Sum and Difference Formulas

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, *sum* formula for sine.

To obtain the identity for

In conclusion, *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

**Solution:** To find the exact value of

Use

For

For

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

**Solutions:**

1.

2.

3.

If

\begin{align*}\sin (y + z) & = \sin y \cos z + \cos y \sin z \\ & = - \frac{5}{13} \cdot - \frac{3}{5} + - \frac{12}{13} \cdot \frac{4}{5} = \frac{15}{65} - \frac{48}{65} = - \frac{33}{65}\end{align*}}}

### Concept Problem Solution

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

\begin{align*} \sin \frac{7\pi}{12} = \sin \left( \frac{4\pi}{12} + \frac{3\pi}{12}\right)\\ =\sin \left( \frac{\pi}{3} + \frac{\pi}{4}\right)\\ =\sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) + cos(\frac{\pi}{3})sin(\frac{\pi}{4})\\ =\left(\frac{\sqrt{3}}{2} \right) \left(\frac{\sqrt{2}}{2} \right) + \left(\frac{1}{2} \right) \left(\frac{\sqrt{2}}{2} \right)\\ =\frac{\sqrt{6}}{4} +\frac{\sqrt{2}}{4}\\ =\frac{\sqrt{6}+ \sqrt{2}}{4}\\ \end{align*}

### Practice

Find the exact value for each sine expression.

- \begin{align*}\sin75^\circ\end{align*}
- \begin{align*}\sin105^\circ\end{align*}
- \begin{align*}\sin165^\circ\end{align*}
- \begin{align*}\sin255^\circ\end{align*}
- \begin{align*}\sin-15^\circ\end{align*}

Write each expression as the sine of an angle.

- \begin{align*}\sin46^\circ\cos20^\circ+\cos46^\circ\sin20^\circ\end{align*}
- \begin{align*}\sin3x\cos2x-\cos3x\sin2x\end{align*}
- \begin{align*}\sin54^\circ\cos12^\circ+\cos54^\circ\sin12^\circ\end{align*}
- \begin{align*}\sin29^\circ\cos10^\circ-\cos29^\circ\sin10^\circ\end{align*}
- \begin{align*}\sin4y\cos3y+\cos4y\sin2y\end{align*}
- Prove that \begin{align*}\sin(x-\frac{\pi}{2})=-\cos(x)\end{align*}
- Suppose that x, y, and z are the three angles of a triangle. Prove that \begin{align*}\sin(x+y)=\sin(z)\end{align*}
- Prove that \begin{align*}\sin(\frac{\pi}{2}-x)=\cos(x)\end{align*}
- Prove that \begin{align*}\sin(x+\pi)=-\sin(x)\end{align*}
- Prove that \begin{align*}\sin(x-y)+\sin(x+y)=2\sin(x)\cos(y)\end{align*}

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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.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.### Image Attributions

Here you'll learn to rewrite sine functions with addition or subtraction in their arguments in a more easily solvable form.