Trigonometric Functions of Associated Angles
Associated angles are angles, usually measured in radians, whose trigonometric functions have the same absolute value. They are also known as associated arcs.
Associated angles are particularly useful because they allow many trigonometric calculations to be simplified. By replacing an angle with one of its associated angles, it is often possible to work with a more familiar angle while obtaining the same trigonometric value, up to its sign.
A practical example
Consider the angles α and -α. Their sine values have the same absolute value:
$$ \sin(\alpha) = |\sin(-\alpha)| $$
For this reason, α and -α are associated angles for the sine function.
Since sine is an odd function, meaning that \(f(-x)=-f(x)\), the sine of -α can be found directly from the sine of α:
$$ \sin(-\alpha) = -\sin(\alpha) $$
Note: The angles α and -α are also associated angles for the cosine function: $$ \cos(\alpha) = |\cos(-\alpha)| $$ In this case, the two angles actually have the same cosine value.
Since cosine is an even function, meaning that \(f(-x)=f(x)\), we have: $$ \cos(-\alpha) = \cos(\alpha) $$
Why are associated angles useful?
One of the main applications of associated angles is the reduction of trigonometric functions to the first quadrant.
Instead of working with an angle located in the second, third, or fourth quadrant, we can often replace it with an associated angle in the first quadrant and then determine the correct sign of the result.
This approach makes trigonometric calculations faster and easier to manage.
Example: Suppose we want to evaluate: $$ \cos(-20^\circ) $$ Since -20° lies in the fourth quadrant and cosine has the same value for opposite angles, we can replace it with its associated angle in the first quadrant: $$ \cos(-20^\circ) = \cos(20^\circ) $$ This allows us to work directly with a first-quadrant angle, which is often more convenient.
Main Formulas for Associated Angles
The following identities are the most commonly used formulas for working with associated angles in trigonometry:
| Associated Angles | Formulas | Proof |
|---|---|---|
| $$ \alpha \ \ , \ \ -\alpha $$ | $$ \sin(-\alpha) = -\sin(\alpha) $$ $$ \cos(-\alpha) = \cos(\alpha) $$ $$ \tan(-\alpha) = -\tan(\alpha) $$ $$ \cot(-\alpha) = -\cot(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ \pi + \alpha $$ | $$ \sin(\pi+\alpha) = -\sin(\alpha) $$ $$ \cos(\pi+\alpha) = -\cos(\alpha) $$ $$ \tan(\pi+\alpha) = \tan(\alpha) $$ $$ \cot(\pi+\alpha) = \cot(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ \pi - \alpha $$ | $$ \sin(\pi-\alpha) = \sin(\alpha) $$ $$ \cos(\pi-\alpha) = -\cos(\alpha) $$ $$ -\tan(\pi-\alpha) = \tan(\alpha) $$ $$ -\cot(\pi-\alpha) = \cot(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ 2\pi - \alpha $$ | $$ \sin(2\pi-\alpha) = -\sin(\alpha) $$ $$ \cos(2\pi-\alpha) = \cos(\alpha) $$ $$ -\tan(2\pi-\alpha) = \tan(\alpha) $$ $$ -\cot(2\pi-\alpha) = \cot(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ \frac{\pi}{2} + \alpha $$ | $$ \sin(\frac{\pi}{2} + \alpha) = \cos(\alpha) $$ $$ \cos(\frac{\pi}{2} + \alpha) = -\sin(\alpha) $$ $$ \tan(\frac{\pi}{2} + \alpha) = -\cot(\alpha) $$ $$ \cot(\frac{\pi}{2} + \alpha) = -\tan(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ \frac{\pi}{2} - \alpha $$ | $$ \sin(\frac{\pi}{2} - \alpha) = \cos(\alpha) $$ $$ \cos(\frac{\pi}{2} - \alpha) = \sin(\alpha) $$ $$ \tan(\frac{\pi}{2} - \alpha) = \cot(\alpha) $$ $$ \cot(\frac{\pi}{2} - \alpha) = \tan(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ \frac{3\pi}{2} + \alpha $$ | $$ \sin(\frac{3\pi}{2} + \alpha) = -\cos(\alpha) $$ $$ \cos(\frac{3\pi}{2} + \alpha) = \sin(\alpha) $$ $$ \tan(\frac{3\pi}{2} + \alpha) = -\cot(\alpha) $$ $$ \cot(\frac{3\pi}{2} + \alpha) = -\tan(\alpha) $$ | see explanation |
| $$ \alpha \ \ , \ \ \frac{3\pi}{2} - \alpha $$ | $$ \sin(\frac{3\pi}{2} - \alpha) = -\cos(\alpha) $$ $$ \cos(\frac{3\pi}{2} - \alpha) = -\sin(\alpha) $$ $$ \tan(\frac{3\pi}{2} - \alpha) = \cot(\alpha) $$ $$ \cot(\frac{3\pi}{2} - \alpha) = \tan(\alpha) $$ | see explanation |
And so on.
