Easy way to remember sin, cos and tan values

One of my tutees showed me an easy way to remember sin and cos values for the “common” angles, and once you’ve got those you’ve also got tan, cosec, sec and cot.

Quite often people need to know (or at least students for exams do!) the \(\sin\) and \(\cos\) values of the most common angles: \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\) (or 0, 30, 45, 60, 90 degrees). Fortunately it is remarkable easy to work out.

First, write down the angles we are looking for

\begin{array}{ccccc}
& 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\
& 0 & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\\hline
\end{array}

Then to work out the angles for \(\sin\) we write down the digits 0 to 4:

\begin{array}{ccccc}
& 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\
& 0 & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\\hline
\sin & 0 & 1 & 2 & 3 & 4
\end{array}

Then take the square root of each of those numbers:

\begin{array}{ccccc}
& 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\
& 0 & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\\hline
\sin & \sqrt{0} & \sqrt{1} & \sqrt{2} & \sqrt{3} & \sqrt{4}
\end{array}

Then divide by 2 and we’re done:

\begin{array}{ccccc}
& 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\
& 0 & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\\hline
\sin & \frac{\sqrt{0}}{2} & \frac{\sqrt{1}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{4}}{2}
\end{array}

To make these the more ‘normal’ values we simplify terms:

\begin{array}{ccccc}
& 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\
& 0 & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\\hline
\sin & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1
\end{array}

Amazing! To find the \(\cos\) terms we do the same, but write the numbers from 4 down to 0. Then square root each, and divide by 2:

\begin{array}{ccccc}
& 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\
& 0 & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\\hline
\sin & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1 \\
\cos & 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & 0
\end{array}

Now using these and our simple trig rules we can find all the other values. The rules to remember are: \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)},\quad\sec(\theta)=\frac{1}{\cos(\theta)},\quad cosec(\theta)=\frac{1}{\sin(\theta)},\quad\cot(\theta)=\frac{1}{\tan(\theta)}\]

Hopefully you’ll never struggle to remember these values again!