Table of values of trigonometric functions of selected angles
Table shows values of trigonometric functions of selected angles. Included functions are: sine, cosine, tangens and cotangens. Both deegres and radians angles are presented.

# Trigonometric functions

 Angle in radians Angle in degrees sin cos tan cot $0$ $0 ^\circ$ $0$ $1$ $0$ $-$ $\frac{\pi}{12}$ $15 ^\circ$ $\frac{\sqrt{6} - \sqrt{2}}{4}$ $\frac{\sqrt{6} + \sqrt{2}}{4}$ $2 - \sqrt{3}$ $2 + \sqrt{3}$ $\frac{\pi}{10}$ $18 ^\circ$ $\frac{\sqrt{5} - 1}{4}$ $\frac{\sqrt{10 + 2\sqrt{5}}}{4}$ $\frac{\sqrt{25 - 10\sqrt{5}}}{5}$ $\sqrt{5 + 2\sqrt{5}}$ $\frac{\pi}{8}$ $22\frac{1}{2} ^\circ$ $\frac{2 - \sqrt{2}}{2}$ $\frac{2 - \sqrt{2}}{2}$ $\sqrt{2} - 1$ $\sqrt{2} + 1$ $\frac{\pi}{6}$ $30 ^\circ$ $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{3}$ $\sqrt{3}$ $\frac{\pi}{4}$ $45 ^\circ$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $1$ $1$ $\frac{\pi}{3}$ $60 ^\circ$ $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ $\frac{\sqrt{3}}{3}$ $\frac{5}{12} \pi$ $75 ^\circ$ $\frac{\sqrt{6} + \sqrt{2}}{4}$ $\frac{\sqrt{6} - \sqrt{2}}{4}$ $2 + \sqrt{3}$ $2 - \sqrt{3}$ $\frac{\pi}{2}$ $90 ^\circ$ $1$ $0$ $-$ $0$

# Some facts

• Trigonometric functions are:
• sine (sinx) - the ratio of the length of the opposite side to the length of the hypotenuse,
$\sin x = \frac{opposite}{hypotenuse}$
• cosine (cosx) - the ratio of the length of the adjacent side to the length of the hypotenuse,
$\cos x = \frac{adjacent}{hypotenuse}$
• tangens (tanx) - the ratio of the length of the side opposite to the angle to the length of the side adjacent to this angle,
$\tan x = \frac{opposite}{adjacent}$
• cotangens (cotx) - the ratio of the length of the side adjacent to the angle to the length of the side that lies opposite this angle,
$\cot x = \frac{adjacent}{opposite}$

• secans (secx) - the ratio of the hypotenuse length to the length of the side adjacent to the angle (inverse cosine),
• cosecans (cosecx) - the ratio of the hypotenuse length to the length of the side opposite the angle of the sine.
• The values of trigonometric functions for frequently used angles can be found in mathematical tables.
• Sometimes we need to find the value of the selected trigonometric function for less typical angles, e.g. a sine of 51 degrees. Then the function's value can be calculated by developing a given function in the so-called Taylor's serie (or more general: power serie).
$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$
$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}$
$\tan x = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots = \sum^{\infty}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}, \quad |x|<\frac{\pi}{2}$
$cot\ x = \frac {1}{x} - \frac {x}{3} - \frac {x^3} {45} - \frac {2 x^5} {945} - \cdots = \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!}, \quad 0 < |x| < \pi$
$sec\ x = 1 + \frac {x^2}{2} + \frac {5 x^4} {24} + \frac {61 x^6} {720} + \cdots = \sum^{\infty}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}, \quad |x|< \frac{\pi}{2}$
• The calculation of the function value by expanding into a power series is used by computers or pocket calculators.

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