Trigonometry

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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.) sum and difference identity 1
2.) sum and difference identity 2
3.) sum and difference identity 3
4.) sum and difference identity 4
5.) sum and difference identity 5
6.) sum and difference identity 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.

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