Trigonometric functions are used to relate the angles of a right triangle to its sides.

,,\qquad \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}\quad\cos(\text{angle}) =\frac{\text{adjacent}}{\text{hypotenuse}}\quad\tan(\text{angle})  = \frac{\sin(\text{angle})}{\cos(\text{angle})} = \frac{\text{opposite}}{\text{adjacent}}\quad

With respect to any of the two angles \[\alpha\] or \[\beta\] (the third angle of a right triangle is, of course, \[90^\text{o}\] thereofre it's irrelevant) three sides can be identified with the following names: \textbf{hypotenuse}, \textbf{opposite} and \textbf{adjacent}. The hypotenuse is easy to identify and it's the longest i.e. \[c\]. The adjacent side is the one that's "touching" the angle; for angle \[\alpha\] that would be \[b\], while for angle \[\beta\] that would be \[a\]. The opposite side is the one that's NOT  "touching" the angle; for angle \[\alpha\] that would be \[a\], while for angle \[\beta\] that would be \[b\].

Therefore, the complete set of trigonometric functions for the triangle in the pic related is:

,,\qquad \sin(\alpha) = \frac{a}{c}\quad \cos(\alpha) = \frac{b}{c}\quad \tan(\alpha) = \frac{a}{b}

,,\qquad \sin(\beta) = \frac{b}{c}\quad \cos(\beta) = \frac{a}{c}\quad \tan(\beta) = \frac{b}{a}


Thus, just knowing the angle \[\alpha\] allows conversion of any side to any other side:
\begin{tabular*}
\{\} & \[a\] &\[b\] & \[c\] \\
\[a=\] & \[a\] & \[b\tan(\alpha)\] & \[c\sin(\alpha)\]\\
\[b=\] & \[\frac{a}{\tan(\alpha)}\] & \[b\] & \[c\cos(\alpha)\]\\
\[c=\] & \[\frac{a}{\sin(\alpha)}\] & \[\frac{b}{\cos(\alpha)}\] & \[c\]
\end{tabular*}
And similarly, just knowing the angle \[\beta\] also allows conversion of any side to any other side:
\begin{tabular*}
\{\} & \[a\] &\[b\] & \[c\] \\
\[a=\] & \[a\] & \[\frac{b}{\tan(\beta)}\] & \[c\cos(\beta)\]\\
\[b=\] & \[a\tan(\beta)\] & \[b\] & \[c\sin(\beta)\]\\
\[c=\] & \[\frac{a}{\cos(\beta)}\] & \[\frac{b}{\sin(\beta)}\] & \[c\]
\end{tabular*}
Intuition behind this is straightforward: For angles of a triangle, sine and cosine are bounded functions between zero and one. \[(0 \leq \sin(x)\leq 1,\; 0\leq \cos(x) \leq1)\]. Multiplying something with these functions should produce a smaller or equal value, while dividing something with these functions should produce a larger or equal value. Since hypotenuse is the longest, multiplying it with sine or cosine will "reduce" it to a leg. Similarly, dividing a leg by sine or cosine will "grow" it into a hypotenuse. A leg can be converted to another leg by "growing" it into a hypotenuse first, then "reducing" it to another leg which effectively is the same as multiplying the leg with a tangent value of the angle that the leg opposes.