Trigonometry - sin, cos, tan, cot

Take an x-axis and an y-axis (orthonormal) and let O be the origin. A circle centered in O and with radius = 1 is known as trigonometric circle or unit circle.
unit triangle
If P is a point from the circle and t is the angle between PO and x then:
  • the x-coordinate of P is called the cosine of t. We write cos(t);
  • the y-coordinate of P is called the sine of t. We write sin(t);
  • the number sin(t)/cos(t) is called the tangent of t. We write tan(t);
  • the number cos(t)/sin(t) is called the cotangent of t. We write cot(t).

The sine function

sin : R -> R
All trigonometric functions are periodic. The period of sin is .
The range of the function is [-1,1].

sin graph

The cosine function

cos : R -> R
The period of sin is .
The range of the function is [-1,1].

cos graph

The tangent function

tan : R -> R
The range of the function is R. Now, the period is π and the function is undefined at x = (π/2) + kπ, k=0,1,2,...
The graph of the tangent function on the interval 0 - π

tan graph

Animated graph(open in a new window):
The graph of the tangent function on the interval 0 - 2π

The cotangent function

cot : R -> R
The range of the function is R. The period is π and that the function is undefined at x = kπ, k=0,1,2,...

cot graph

The values of sin, cos, tan, cot at the angles of 0°, 30°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°

$\alpha^o$ $0^o$ $30^o$ $45^o$ $60^o$ $90^o$ $120^o$ $135^o$ $150^o$ $180^o$ $210^o$ $225^o$ $240^o$ $270^o$ $300^o$ $315^o$ $330^o$ $360^o$
$\alpha rad$ $0$ $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$ $\frac{2\pi}{3}$ $\frac{3\pi}{4}$ $\frac{5\pi}{6}$ $\pi$ $\frac{7\pi}{6}$ $\frac{5\pi}{4}$ $\frac{4\pi}{3}$ $\frac{3\pi}{2}$ $\frac{5\pi}{3}$ $\frac{7\pi}{4}$ $\frac{11\pi}{6}$ $2\pi$
$sin\alpha$ $0$ $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{3}}{2}$ $-1$ $-\frac{\sqrt{3}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{1}{2}$ $0$
$cos\alpha$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{3}}{2}$ $-1$ $-\frac{\sqrt{3}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{1}{2}$ $0$ $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ $1$
$tan\alpha$ $0$ $\frac{\sqrt{3}}{3}$ $1$ $\sqrt{3}$ $-$ $-\sqrt{3}$ $-1$ $-\frac{\sqrt{3}}{3}$ $0$ $\frac{\sqrt{3}}{3}$ $1$ $\sqrt{3}$ $-$ $-\sqrt{3}$ $-1$ $-\frac{\sqrt{3}}{3}$ $0$
$cot\alpha$ $-$ $\sqrt{3}$ $1$ $\frac{\sqrt{3}}{3}$ $0$ $-\frac{\sqrt{3}}{3}$ $-1$ $-\sqrt{3}$ $-$ $\sqrt{3}$ $1$ $\frac{\sqrt{3}}{3}$ $0$ $-\frac{\sqrt{3}}{3}$ $-1$ $-\sqrt{3}$ $-$

The easiest way to remember the basic values of sin and cos at the angles of 0°, 30°, 60°, 90°:
sin([0, 30, 45, 60, 90]) = cos([90, 60, 45, 30, 0]) = sqrt([0, 1, 2, 3, 4]/4)

Trigonometric identities

With t radians corresponds exactly one point P(cos(t),sin(t)) on the unit circle. The square of the distance [OP] = 1. Calculating this distance with the coordinates of P we have for each t:

cos2(t) + sin2(t) = 1

If t + t' = 180° then:

  • sin(t) = sin(t')
  • cos(t) = -cos(t')
  • tan(t) = -tan(t')
  • cot(t) = -cot(t')

If t + t' = 90° then:

  • sin(t) = cos(t')
  • cos(t) = sin(t')
  • tan(t) = cot(t')
  • cot(t) = tan(t')

$-\alpha$ $90^\circ - \alpha$ $90^\circ + \alpha$ $180^\circ - \alpha$
$\textrm{ sin }$ $-\textrm{ sin }\alpha$ $\textrm{ cos }\alpha$ $\textrm{ cos } \alpha$ $\textrm{ sin }\alpha$
$\textrm{ cos }$ $\textrm{ cos }\alpha$ $\textrm{ sin }\alpha$ $-\textrm{ sin} \alpha$ $-\textrm{ cos }\alpha$
$\textrm{ tan }$ $-\textrm{ tan }\alpha$ $\textrm{ cot }\alpha$ $-\textrm{ cot } \alpha$ $-\textrm{ tan }\alpha$
$\textrm{ cot }$ $-\textrm{ cot }\alpha$ $\textrm{ tan }\alpha$ $-\textrm{ tan } \alpha$ $-\textrm{ cot }\alpha$

Trigonometric formulas

$\cos(u - v) = \cos(u)\cdot \cos(v) + \sin(u)\cdot \sin(v)$

$\cos(u + v) = \cos(u - (-v)) = \cos(u)\cdot \cos(-v) + \sin(u)\cdot \sin(-v)$

$\sin(u - v) = \sin(u)\cdot \cos(v) - \cos(u)\cdot \sin(v)$

$\sin(u + v) = \sin(u)\cdot \cos(v) + \cos(u)\cdot \sin(v)$

$\tan(u + v) = \frac{sin(u + v)}{\cos(u + v)}=\frac{\sin(u)\cdot \cos(v) + \cos(u)\cdot \sin(v)}{\cos(u)\cdot \cos(v) - \sin(u)\cdot \sin(v)}$

$\tan(u + v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u).tan(v)}$


$\sin(2u) = 2\sin(u)\cdot \cos(u)$

$\cos(2u) = \cos^2(u) - \sin^2(u) = 2\cos^2(u) - 1 = 1 - 2sin^2(u)$

$\tan(2u) = \frac{2\tan(u)}{1- \tan^2(u)}$

$\cos(2u) = \frac{1 - \tan^2(u)}{1 + \tan^2(u)}$

$\sin(2u) = \frac{2\tan(u)}{1 + \tan^2(u)}$

$1 + \cos(2u) = 2 \cos^2(u)$

$1 - \cos(2u) = 2 \sin^2(u)$


$\sin3\beta = 3\sin\beta - 4 sin^3\beta$

$\cos3\beta = 4\cos^3\beta - 4 cos\beta$

$\tan3\beta=\frac{3\tan\beta - \tan^3\beta}{1-3\tan^2\beta}$

$\cot3\beta=\frac{\cot^3\beta-3\cot\beta}{3\cot^2\beta-1}$

Addition and multiplication of sin and cos

$\textrm{ sin } \alpha + \textrm{ sin }\beta = 2 \textrm{ sin }\frac{\alpha + \beta}{2} \textrm{ cos }\frac{\alpha - \beta}{2}$

$\textrm{ sin } \alpha - \textrm{ sin }\beta = 2 \textrm{ sin }\frac{\alpha - \beta}{2} \textrm{ cos }\frac{\alpha + \beta}{2}$

$\textrm{ cos } \alpha + \textrm{ cos }\beta = 2 \textrm{ cos }\frac{\alpha + \beta}{2} \textrm{ cos }\frac{\alpha - \beta}{2}$

$\textrm{ cos } \alpha - \textrm{ cos }\beta = -2 \textrm{ sin }\frac{\alpha + \beta}{2} \textrm{ sin }\frac{\alpha - \beta}{2}$

$\textrm{ sin }\alpha \textrm{ sin }\beta = \frac{1}{2} (\textrm{ cos }(\alpha - \beta) - \textrm{ cos }(\alpha + \beta))$

$\textrm{ cos }\alpha \textrm{ cos }\beta = \frac{1}{2} (\textrm{ cos }(\alpha - \beta) + \textrm{ cos }(\alpha + \beta))$

$\textrm{ sin }\alpha \textrm{ cos }\beta = \frac{1}{2} (\textrm{ sin }(\alpha + \beta) + \textrm{ sin }(\alpha - \beta))$



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