Consider a right triangle $ABC$ with the right angle at vertex $C$, side of length $a$ opposite vertex $A$, with $ \angle BAC$ of measure $\alpha$, and side of length $b$ opposite vertex $B$, with $ \angle ABC $ of measure $\beta$, and hypotenuse of length $c$.

From the geometry of similar triangles applied to this right triangle, once $\alpha$ is determined, the ratio of any specified pair of sides is constant, depending only on the pair chosen and not the magnitudes of the sides.

Function Name |
Abbreviation for Function Name |
Function Value for $\alpha$ |

sine |
sin |
$\sin(\alpha) = \frac ac$ |

cosine |
cos |
$\cos(\alpha) = \frac bc$ |

tangent |
tan |
$\tan(\alpha) = \frac ab$ |

cotangent |
cot |
$\cot(\alpha) = \frac ba$ |

secant |
sec |
$\sec(\alpha) = \frac cb$ |

cosecant |
csc |
$\csc(\alpha) = \frac ca$ |

We start by considering two special triangles, the 30-60-90 and 45-45-90 triangles.

$\Theta$ |
$0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |

$\sin(\Theta)$ |
$0$ | $\frac 12$ |
$\frac 1{\sqrt{2}} = \frac {\sqrt{2}}2$ | $\frac {\sqrt{3}}2$ | $1$ |

$\cos(\Theta)$ | $1$ | $\frac {\sqrt{3}}2$ | $\frac 1{\sqrt{2}} = \frac {\sqrt{2}}2$ | $\frac 12$ | $0$ |

$\tan(\Theta)$ | $0$ | $\frac 1{\sqrt{3}} = \frac {\sqrt{3}}3$ | $1$ |
$\sqrt{3}$ | Undefined |

$\cot(\Theta)$ | Undefined |
$\sqrt{3}$ | $1$ |
$\frac 1{\sqrt{3}} = \frac {\sqrt{3}}3$ | $0$ |

$\sec(\Theta)$ | $1$ | $\frac 2{\sqrt{3}}= \frac {2\sqrt{3}}3$ | $\sqrt{2}$ | $2$ | Undefined |

$\csc(\Theta)$ | Undefined |
$2$ | $\sqrt{2}$ | $\frac 2{\sqrt{3}}= \frac {2\sqrt{3}}3$ | $1$ |

Here are two dynamic calculators created with GeoGebra visualizing the triangle trigonometric
functions with triangles with $b =1$ (in red) and triangles with $c=1$ (in blue).
The first calculates all six function values while the second focuses
on the sine, cosine and tangent functions along with
their mapping diagrams using degree measure for the domain axis.