Trigonometric Function Cosine

The cosine function

The cosine function is very similar to the sine function. It just uses different leg in it’s equation. Thus the definition is: For the angle α, the cosine function gives the ratio of the length of the adjacent side to the length of the hypotenuse. For our triangle, it holds:

\Large\sin(\alpha)=\frac{\mbox{length of adjacent side}}{\mbox{length of hypotenuse}}=\frac{|AB|}{|BC|}

As you can see, the cosine function just uses the adjacent side instead of opposite side in the formula. We can simply compute the length of the AB side using the same approach. We adjust the equation such that the only unknown variable |AB| is on the left side:

\begin{eqnarray}\cos(\alpha)&=&\frac{|AB|}{|BC|}\qquad /\cdot |BC|\\|BC|\cdot\cos(\alpha)&=&|AB|\\|AB|&=&|BC|\cdot\cos(\alpha)\end{eqnarray}

We can use the Google again to compute the value of cos(30°) and we see it’s equals to circa 0.866). Not a nice number like before but that’s life. We can substitute known values

|AB| = |BC| · cos(α) = 6 · 0.866 = 5.196.

This is good enough result, but we can do better. The value cos(30°) is actually a very known value since

\cos(30 ^\circ)=\frac{\sqrt{3}}{2}.

Using this knowledge we can express the result more precisely as

|AB|=|BC|\cdot\cos(\alpha)=6\cdot\frac{\sqrt{3}}{2} = 3\cdot\sqrt{3}.

Known special values of sine and cosine

\LARGE\begin{matrix}&\sin&\cos\\0^\circ&0&1\\30^\circ&\frac12&\frac{\sqrt{3}}{2}\\45^\circ&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\\60^\circ&\frac{\sqrt{3}}{2}&\frac12\\90^\circ&1&0\end{matrix}