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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 radiansAngle in degreessincostancot
0000 ^\circ001100-
π12\frac{\pi}{12}1515 ^\circ624\frac{\sqrt{6} - \sqrt{2}}{4}6+24\frac{\sqrt{6} + \sqrt{2}}{4}232 - \sqrt{3}2+32 + \sqrt{3}
π10\frac{\pi}{10}1818 ^\circ514\frac{\sqrt{5} - 1}{4}10+254\frac{\sqrt{10 + 2\sqrt{5}}}{4}251055\frac{\sqrt{25 - 10\sqrt{5}}}{5}5+25\sqrt{5 + 2\sqrt{5}}
π8\frac{\pi}{8}221222\frac{1}{2} ^\circ222\frac{2 - \sqrt{2}}{2}222\frac{2 - \sqrt{2}}{2}21\sqrt{2} - 12+1\sqrt{2} + 1
π6\frac{\pi}{6}3030 ^\circ12\frac{1}{2}32\frac{\sqrt{3}}{2}33\frac{\sqrt{3}}{3}3\sqrt{3}
π4\frac{\pi}{4}4545 ^\circ22\frac{\sqrt{2}}{2}22\frac{\sqrt{2}}{2}1111
π3\frac{\pi}{3}6060 ^\circ32\frac{\sqrt{3}}{2}12\frac{1}{2}3\sqrt{3}33\frac{\sqrt{3}}{3}
512π\frac{5}{12} \pi7575 ^\circ6+24\frac{\sqrt{6} + \sqrt{2}}{4}624\frac{\sqrt{6} - \sqrt{2}}{4}2+32 + \sqrt{3}232 - \sqrt{3}
π2\frac{\pi}{2}9090 ^\circ1100-00

Some facts

  • Trigonometric functions are:
    • sine (sinx) - the ratio of the length of the opposite side to the length of the hypotenuse,
      sinx=oppositehypotenuse\sin x = \frac{opposite}{hypotenuse}
    • cosine (cosx) - the ratio of the length of the adjacent side to the length of the hypotenuse,
      cosx=adjacenthypotenuse\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,
      tanx=oppositeadjacent\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,
      cotx=adjacentopposite\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).
    sinx=xx33!+x55!x77!+=n=0(1)nx2n+1(2n+1)!\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)!}
    cosx=1x22!+x44!x66!+=n=0(1)nx2n(2n)!\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)!}
    tanx=x+x33+2x515+=n=1B2n(4)n(14n)(2n)!x2n1,x<π2\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=1xx3x3452x5945=n=0(1)n22nB2nx2n1(2n)!,0<x<πcot\ 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+x22+5x424+61x6720+=n=0(1)nE2n(2n)!x2n,x<π2sec\ 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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