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 circle
If P is a point from the circle and A is the angle between PO and x axis then:
  • the x-coordinate of P is called the cosine of A. We write cos(A) or cos A;
  • the y-coordinate of P is called the sine of A. We write sin(A) or sin A;
  • the number sin(A)/cos(A) is called the tangent of A. We write tan(A) or tan A;
  • the number cos(A)/sin(A) is called the cotangent of A. We write cot(A) or cot A.

The sine function

sin : R -> R
All trigonometric functions are periodic. The period of sin is 2$\pi$.
The range of the function is [-1,1].

sin graph

The cosine function

cos : R -> R
The period of sin is 2$\pi$.
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 $\pi$ and the function is undefined at x = ($\pi$/2) + k$\pi$, k=0,1,2,...
The graph of the tangent function on the interval 0 - $\pi$

tan graph

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

The cotangent function

cot : R -> R
The range of the function is R. The period is $\pi$ and that the function is undefined at x = k$\pi$, 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°

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

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)

Basic Trigonometric Identities

For every angle A corresponds exactly one point P(cos(A),sin(A)) on the unit circle.

cos2(A) + sin2(A) = 1

If A + B = 180° then:

  • sin(A) = sin(B)
  • cos(A) = -cos(B)
  • tan(A) = -tan(B)
  • cot(A) = -cot(B)

If A + B = 90° then:

  • sin(A) = cos(B)
  • cos(A) = sin(B)
  • tan(A) = cot(B)
  • cot(A) = tan(B)

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

Trigonometric Formulas

Half-Angle Formulas

$\sin\frac{A}{2}=\pm\sqrt{\frac{1-\cos A}{2}}$
+ if $\frac{A}{2}$ lies in quadrant | or ||
- if $\frac{A}{2}$ lies in quadrant ||| or |V


$\cos\frac{A}{2}=\pm\sqrt{\frac{1+\cos A}{2}}$
+ if $\frac{A}{2}$ lies in quadrant | or |V
- if $\frac{A}{2}$ lies in quadrant || or |||


$\tan\frac{A}{2}=\pm\sqrt{\frac{1-\cos A}{1+\cos A}}$
+ if $\frac{A}{2}$ lies in quadrant | or |||
- if $\frac{A}{2}$ lies in quadrant || or |V


$\cot\frac{A}{2}=\pm\sqrt{\frac{1+\cos A}{1-\cos A}}$
+ if $\frac{A}{2}$ lies in quadrant | or |||
- if $\frac{A}{2}$ lies in quadrant || or |V


$\tan\frac{A}{2} = \frac{\sin A}{1+\cos A} = \frac{1-\cos A}{\sin A}=\csc A-\cot A$

$\cot\frac{A}{2} = \frac{\sin A}{1-\cos A} = \frac{1+\cos A}{\sin A}=\csc A+\cot A$

Double and Triple Angle Formulas

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

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

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

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

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


$\sin3A = 3\sin A - 4 \sin^3A$

$\cos3A = 4\cos^3A - 4 \cos A$

$\tan3A=\frac{3\tan A - \tan^3A}{1-3\tan^2A}$

$\cot3A=\frac{\cot^3A-3\cot A}{3\cot^2A-1}$


$\sin4A = 4\cos^3A\cdot \sin A - 4\cos A\cdot \sin^3A$

$\cos4A = \cos^4A - 6\cos^2A\cdot \sin^2A + \sin^4A$

$\tan4A=\frac{4\tan A - 4\tan^3A}{1-6\tan^2A+\tan^4A}$

$\cot4A=\frac{\cot^4A-6\cot^2A+1}{4\cot^3A-4\cot A}$

Power-Reducing Formulas

$\sin^2(A)=\frac{1 - \cos(2A)}{2}$

$\sin^3(A)=\frac{3\sin A - \sin(3A)}{4}$

$\sin^4(A)=\frac{\cos(4A) - 4\cos(2A) + 3}{8}$


$\cos^2(A) = \frac{1 + \cos(2A)}{2}$

$\cos^3(A)=\frac{3\cos A + \cos(3A)}{4}$

$\cos^4(A)=\frac{4\cos(2A) + \cos(4A) + 3}{8}$

Sum and Difference of Angles

$\sin(A + B) = \sin(A)\cdot \cos(B) + \cos(A)\cdot \sin(B)$

$\sin(A - B) = \sin(A)\cdot \cos(B) - \cos(A)\cdot \sin(B)$


$\cos(A + B) = \cos(A)\cdot \cos(B) - \sin(A)\cdot \sin(B)$

$\cos(A - B) = \cos(A)\cdot \cos(B) + \sin(A)\cdot \sin(B)$


$\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)}=\frac{\sin(A)\cdot \cos(B) + \cos(A)\cdot \sin(B)}{\cos(A)\cdot \cos(B) - \sin(A)\cdot \sin(B)}$

$\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\cdot\tan(B)}$

$\cot(A \pm B) = \frac{\cot(B)\cot(A)\mp 1}{\cot(B)\pm \cot(A)}=\frac{1\mp \tan(A)\tan(B)}{\tan(A)\pm \tan(B)}$


$\sin(A + B + C) = \sin A\cdot\cos B\cdot\cos C + \cos A\cdot\sin B\cdot\cos C + \cos A\cdot\cos B\cdot\sin C - \sin A\cdot\sin B\cdot\sin C$

$\cos(A + B + C) = \cos A\cdot\cos B\cdot\cos C - \sin A\cdot\sin B\cdot\cos C - \sin A\cdot\cos B\cdot\sin C $
$- \sin A\cdot\cos B \cdot\sin C - \cos A \cdot \sin B\cdot \sin C$

$\tan(A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A\cdot \tan B \cdot \tan C}{1 - \tan A \cdot\tan B - \tan B\cdot\tan C - \tan A\cdot\tan C}$

Sum and Difference of Trigonometric Functions

$\textrm{ sin } A + \textrm{ sin }B = 2 \textrm{ sin }\frac{A + B}{2} \textrm{ cos }\frac{A - B}{2}$

$\textrm{ sin } A - \textrm{ sin }B = 2 \textrm{ sin }\frac{A - B}{2} \textrm{ cos }\frac{A + B}{2}$


$\textrm{ cos } A + \textrm{ cos }B = 2 \textrm{ cos }\frac{A + B}{2} \textrm{ cos }\frac{A - B}{2}$

$\textrm{ cos } A - \textrm{ cos }B = -2 \textrm{ sin }\frac{A + B}{2} \textrm{ sin }\frac{A - B}{2}$


$\tan A + \tan B = \frac{\sin(A+B)}{\cos A \cdot\cos B}$

$\tan A - \tan B = \frac{\sin(A-B)}{\cos A\cdot\cos B}$


$\cot A + \cot B = \frac{\sin(A+B)}{\sin A\cdot\sin B}$

$\cot A - \cot B = \frac{-\sin(A-B)}{\sin A\cdot\sin B}$

Multiplication of 2 Trigonometric Functions

$\textrm{ sin }A \textrm{ sin }B = \frac{1}{2} (\textrm{ cos }(A - B) - \textrm{ cos }(A + B))$

$\textrm{ cos }A \textrm{ cos }B = \frac{1}{2} (\textrm{ cos }(A - B) + \textrm{ cos }(A + B))$

$\textrm{ sin }A \textrm{ cos }B = \frac{1}{2} (\textrm{ sin }(A + B) + \textrm{ sin }(A - B))$


$\tan A \cdot \tan B = \frac{\tan A+\tan B}{\cot A+\cot B}=-\frac{\tan A-\tan B}{\cot A-\cot B}$

$\cot A \cdot \cot B = \frac{\cot A+\cot B}{\tan A+\tan B}$

$\tan A \cdot \cot B = \frac{\tan A+\cot B}{\cot A+\tan B}$


$\sin A\sin B\sin C = \frac{1}{4}\big(\sin(A+B-C)+\sin(B+C-A)+\sin(C+A-B)-\sin(A+B+C)\big)$

$\cos A\cos B\cos C = \frac{1}{4}\big(\cos(A+B-C)+\cos(B+C-A)+\cos(C+A-B)+\cos(A+B+C)\big)$

$\sin A\sin B\cos C = \frac{1}{4}\big(-\cos(A+B-C)+\cos(B+C-A)+\cos(C+A-B)-\cos(A+B+C)\big)$

$\sin A\cos B\cos C = \frac{1}{4}\big(\sin(A+B-C)-\sin(B+C-A)+\sin(C+A-B)+\sin(A+B+C)\big)$

Tangent half-angle substitution

$\sin A = \frac{2\tan\frac{A}{2}}{1+\tan^2\frac{A}{2}}$

$\cos A = \frac{1-\tan^2\frac{A}{2}}{1+\tan^2\frac{A}{2}}$

$\tan A = \frac{2\tan\frac{A}{2}}{1-\tan^2\frac{A}{2}}$

$\cot A = \frac{1-\tan^2\frac{A}{2}}{2\tan\frac{A}{2}}$

Other Trigonometric Formulas

$1\pm\sin A=2\sin^2\big(\frac{\pi}{4}\pm \frac{A}{2}\big)=2\cos^2\big(\frac{\pi}{4}\mp \frac{A}{2}\big)$


$\frac{1-\sin A}{1+\sin A} = \tan^2(\frac{\pi}{4}-\frac{A}{2})$

$\frac{1-\cos A}{1+\cos A} = \tan^2\frac{A}{2}$

$\frac{1-\tan A}{1+\tan A} = \tan(\frac{\pi}{4}-A)$

$\frac{1+\tan A}{1-\tan A} = \tan(\frac{\pi}{4}+A)$

$\frac{\cot A + 1}{\cot A - 1} = \cot(\frac{\pi}{4}-A)$

$\tan A + \cot A = \frac{2}{\sin2A}$

$\tan A - \cot A = -2\cot2A$

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