(i) Right Triangle Trigonometric Functions
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.

For $\alpha$ measured in degrees, $0^{\circ} < \alpha  <  90^{\circ} $ or $\alpha$  measured with radians, $0  < \alpha < \frac {\pi}2$, based on the choice of sides, the following ratios define the value of the trigonometric function for $\alpha$:

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.

Here is a table for the trigonometric functions for the related angles measured in degrees and a simple mapping diagram for the sine and cosine functions also using degree measure:
$\Theta$
 $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $60^{\circ}$ $90^{\circ}$
$\sin(\Theta)$
$0$ $\frac 12$
$\frac 1{\sqrt{2}} = \frac {\sqrt{2}}2$ $\frac {\sqrt{3}}2$ $1$
$\cos(\Theta)$ $1$ $\frac {\sqrt{3}}2$ $\frac 1{\sqrt{2}} = \frac {\sqrt{2}}2$ $\frac 12$ $0$
$\tan(\Theta)$ $0$ $\frac 1{\sqrt{3}} = \frac {\sqrt{3}}3$ $1$
$\sqrt{3}$ 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.