Easy way to remember sin, cos and tan values

One of my tutees showed me an easy way to remember sine and cosine values for the common angles. Once you’ve got those, you can also determine tangent, cosecant, secant, and cotangent values.

Often, students need to know the sine and cosine values of the most common angles: $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$ (or 0°, 30°, 45°, 60°, and 90°). Fortunately, it’s remarkably easy to work these out.

First, write down the angles we are looking for:

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

To determine the sine values, write down the digits 0 to 4:

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

Next, take the square root of each of those numbers:

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

Then, divide each by 2:

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

Simplifying these terms:

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

To find the cosine values, write the numbers from 4 down to 0, then take the square root of each and divide by 2:

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

Using these values and basic trigonometric identities, you can find all other trigonometric functions. The key identities to remember are:

\[\begin{align*} \tan(\theta) &= \frac{\sin(\theta)}{\cos(\theta)} \\ \sec(\theta) &= \frac{1}{\cos(\theta)} \\ \csc(\theta) &= \frac{1}{\sin(\theta)} \\ \cot(\theta) &= \frac{1}{\tan(\theta)} \end{align*}\]

With this approach, remembering these trigonometric values becomes straightforward.