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In calculus classes when you are asked to evaluate a trig function at a specific angle, it’s 99.9% of the time at one of the so-called special angles we use in our chart. Since you are likely to have learned degrees first I’ll include degree angles in the first chart, but after that, it’s going to be radian only.
Begin by setting up the table on scratch paper as follows:
$$\begin{array}{ l| |c|c|c|c|c } \theta & 0 = 0º & \tfrac{\pi}{6} = 30º & \tfrac{\pi}{4}=45º & \tfrac{\pi}{3}=60º & \tfrac{\pi}{2}=90º \\ \hline\hline \sin\theta & & & & & \\ \hline \cos\theta & & & & & \\ \hline \tan\theta & & & & & \\ \hline \end{array} $$
Then remember ##\sin\theta## starts at zero, fill in the pattern
$$\begin{array}{ l| |c|c|c|c|c } \theta & 0 & \tfrac{\pi}{6} & \tfrac{\pi}{4} & \tfrac{\pi}{3} & \tfrac{\pi}{2} \\ \hline\hline\sin\theta & \tfrac{\sqrt{0}}{2} & \tfrac{\sqrt{1}}{2} & \tfrac{\sqrt{2}}{2} & \tfrac{\sqrt{3}}{2} & \tfrac{\sqrt{4}}{2} \\...
##\theta~\rightarrow## | 0° = 0 | 30° = ##\dfrac{\pi}{6}## | 45° = ##\dfrac{\pi}{4}## | 60° = ##\dfrac{\pi}{3}## | 90° = ##\dfrac{\pi}{2}## |
##\sin \theta## | ##\sqrt{\dfrac{0}{4}}## | ##\sqrt{\dfrac{1}{4}}## | ##\sqrt{\dfrac{2}{4}}## | ##\sqrt{\dfrac{3}{4}}## | ##\sqrt{\dfrac{4}{4}}## |
##\cos \theta## | ##\sqrt{\dfrac{4}{4}}## | ##\sqrt{\dfrac{3}{4}}## | ##\sqrt{\dfrac{2}{4}}## | ##\sqrt{\dfrac{1}{4}}## | ##\sqrt{\dfrac{0}{4}}## |
##\tan \theta## | ##\sqrt{\dfrac{0}{4 - 0}}## | ##\sqrt{\dfrac{1}{4 - 1}}## | ##\sqrt{\dfrac{2}{4 - 2}}## | ##\sqrt{\dfrac{3}{4 - 3}}## | ##\sqrt{\dfrac{4}{4 - 4}}## |
The trick involves using a mnemonic device to remember the values of the sine, cosine, and tangent of special angles (0°, 30°, 45°, 60°, and 90°). It is a helpful tool for quickly calculating these values without having to use a calculator.
The mnemonic device is "SOH-CAH-TOA", which stands for "Sine equals Opposite over Hypotenuse", "Cosine equals Adjacent over Hypotenuse", and "Tangent equals Opposite over Adjacent".
Knowing the values of these special angles can be useful in various fields, such as mathematics, engineering, and physics. It can also save time when solving trigonometric equations or problems.
One way to practice is by using flashcards or creating a study guide with the values and their corresponding angles. You can also try solving different types of problems that involve these special angles.
Aside from using the mnemonic device, you can also try creating visual aids, such as diagrams or charts, to help you remember the values. It may also be helpful to understand the relationship between the special angles and the unit circle.