Trigonometric reduction formulas

Angle in degrees	Angle in radians	Sine	Cosine	Tangens	Cotangens
$-\alpha$	$-\alpha$	$-sin(\alpha)$	$cos(\alpha)$	$-tg(\alpha)$	$-ctg(\alpha)$
$90^\circ + \alpha$	$\frac{\pi}{2} + \alpha$	$cos(\alpha)$	$-sin(\alpha)$	$-ctg(\alpha)$	$-tg(\alpha)$
$90^\circ - \alpha$	$\frac{\pi}{2} - \alpha$	$cos(\alpha)$	$sin(\alpha)$	$ctg(\alpha)$	$tg(\alpha)$
$180^\circ + \alpha$	$\pi + \alpha$	$-sin(\alpha)$	$-cos(\alpha)$	$tg(\alpha)$	$ctg(\alpha)$
$180^\circ - \alpha$	$\pi - \alpha$	$sin(\alpha)$	$-cos(\alpha)$	$-tg(\alpha)$	$-ctg(\alpha)$
$270^\circ + \alpha$	$\frac{3}{2}\pi + \alpha$	$-cos(\alpha)$	$sin(\alpha)$	$-ctg(\alpha)$	$-tg(\alpha)$
$270^\circ - \alpha$	$\frac{3}{2}\pi - \alpha$	$-cos(\alpha)$	$-sin(\alpha)$	$ctg(\alpha)$	$tg(\alpha)$
$360^\circ + \alpha$	$2\pi + \alpha$	$sin(\alpha)$	$cos(\alpha)$	$tg(\alpha)$	$ctg(\alpha)$
$360^\circ - \alpha$	$2\pi - \alpha$	$-sin(\alpha)$	$cos(\alpha)$	$-tg(\alpha)$	$-tg(\alpha)$

Some facts

Reduction formulas allow conversion of trigonometric expressions of obtuse angle into equivalent (and simpler) form containing acute angle.
We often prefer expression containing obtuse angle, because math tables contain values of trigonometric functions for these angles.
The basis of all reduction formulas are fact, that trigonometric functions are periodic. It means that their values cyclically repeat every certain angle. This angle is called period.

ⓘ Example: The basic period of sine function is $2\pi$ ( $360^\circ$ ), because:
$sin(\alpha + 2\pi) = sin(\alpha)$

ⓘ Example: The basic period of tangens function is $\pi$ ( $180^\circ$ ), because:
$tg(\alpha + \pi) = tg(\alpha)$

How to use this tool

1. First, convert your angle into one of below forms:
- $\frac{\pi}{2} \pm \alpha$ ,
- $\pi \pm \alpha$ ,
- $\frac{3}{2}\pi \pm \alpha$ ,
- $2\pi \pm \alpha$ .
Or using degrees:
- $90^\circ \pm \alpha$ ,
- $180^\circ \pm \alpha$ ,
- $270^\circ \pm \alpha$ ,
- $360^\circ \pm \alpha$ .
2. Next, find row containing your new angle in reduction formulas table.
3. Finally, find column containing your trigonometric function and replace your expression with one from table.
ⓘ Example: We want to calculate sine of 120 degrees.
- 1. We found, that 120 degrees can be written as:
  $120^\circ = 90^\circ + 30^\circ$
- 2. We see that our new angle matches to below form in reduction formulas table: $90^\circ + \alpha$
- 3. We look at sine column and we finally found:
  $sin(90^\circ + 30^\circ) = cos(30^\circ) = \frac{\sqrt{3}}{2}$
That's all folks!

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