Table of trigonometric reduction formulas
Table shows so-called rediction formulas, which allow to calculate values of trigonometric functions of obtuse angle (more than 90 degrees) without calculator easily.

# Trigonometric reduction formulas

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

# 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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