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:
$tg(\alpha + \pi) = tg(\alpha)$
ⓘ Hint: If you are interested in trigonometry you can checkout our other calculators:
- reduction formulas - so-called reduction formulas table, that help to calculate value of trigonometric functions for less common angles,
- trigonometric functions values - a table containing the values of trigonometric functions for the most common angles, e.g. sin 90 degrees,
- trigonometric identities - a list of different, more or less popular, dependencies between various trigonometric functions.

How to use this tool#

1. First, convert your angle into one of below forms:
- $\dfrac{\pi}{2} \pm \alpha$ ,
- $\pi \pm \alpha$ ,
- $\dfrac{3}{2}\pi \pm \alpha$ ,
- $2\pi \pm \alpha$ .
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:
  $sin(90^\circ + 30^\circ) = cos(30^\circ) = \dfrac{\sqrt{3}}{2}$
That's all folks!

Tags and links to this website#

Tags:

reduction_formulas · trigonometric_reduction_formulas · sine_of_obtuse_angle · cosine_of_obtuse_angle · tangens_of_obtuse_angle · cotangens_of_obtuse_angle · sine_reductinon_formulas · cosine_reductinon_formulas · tangens_reductinon_formulas · cotangens_reductinon_formulas

Tags to Polish version:

wzory_redukcyjne · trygonometryczne_wzory_redukcyjne · sinus_kata_rozwartego · cosinus_kata_rozwartego · tangens_kata_rozwartego · cotangens_kata_rozwartego · wzory_redukcyjne_sinusa · wzory_redukcyjne_cosinusa · wzory_redukcyjne_tangensa · wzory_redukcyjne_cotangensa

What tags this calculator has#

TABLE · MATH · A -> Z (all)

Permalink#

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

https://calculla.com/reduction_formulas

Links to external sites (leaving Calculla?)#