Great news! We will be upgrading our calculator and lesson pages over the next few months. If you notice any issues, you can submit a contact form by clicking here.

Sum and Difference Identities

The Identities

In trigonometry, there are six sum and difference identities. They are useful when the given angle in a trigonometry expression cannot be evaluated. The six sum and difference identities are given as:

1.)
2.)
3.)
4.)
5.)
6.)

Example Problem

Evaluate the following trigonometry expressions by using the sum and difference identities.
1.) $\sin(\frac{\pi}{12})$
2.) $\cos(\frac{\pi}{12})$
3.) $\tan(\frac{\pi}{12})$

Solution:
To evaluate the angle $\frac{\pi}{12}$ without a calculator we can use $\frac{\pi}{4}$ – $\frac{\pi}{6}$ .
This means we use the difference identities where a – b = $\frac{\pi}{4}$ – $\frac{\pi}{6}$ . Therefore, a = $\frac{\pi}{4}$ and b = $\frac{\pi}{6}$ .

1.) $\sin(\frac{\pi}{12}) = \sin ( \frac{\pi}{4} )\cos (\frac{\pi}{6}) - \cos ( \frac{\pi}{4})\sin (\frac{\pi}{6})$
The final answer is $\frac{\sqrt6-\sqrt2}{4}$ or 0.259.

2.) $\cos(\frac{\pi}{12}) = \cos ( \frac{\pi}{4})\cos (\frac{\pi}{6}) + \sin ( \frac{\pi}{4})\sin (\frac{\pi}{6})$
The final answer is $\frac{\sqrt6+\sqrt2}{4}$ or 0.966.

3.) $\tan(\frac{\pi}{12}) = \frac{\tan ( \frac{\pi}{4}) - \tan (\frac{\pi}{6})}{1 + \tan ( \frac{\pi}{4}) \tan (\frac{\pi}{6})}$
The final answer is $2-\sqrt3}$ or 0.268.

Related Content

Sum and Difference Identities

The Identities

Example Problem