How to calculate $\theta = \arcsin\left(\frac{1}{2}\right)$ with given four function calculator?

Question

An example question is:

In radian measure, what is $\arcsin \left(\frac{1}{2}\right)$?

Select one:

a. $0$

b. $\frac{\pi}{6}$

c. $\frac{\pi}{4}$

d. $\frac{\pi}{3}$

e. $\frac{\pi}{2}$

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So, in the exam, I will be given only four function calculator. And is it possible to calculate this kind of trigo function? Or, do I have to memorise common values of trigo functions? Is there any tricks and tips for this problem?

Answer 1

The function $\arcsin$ is the inverse of $\sin$.

So to compute $\arcsin(\frac{1}{2})$ we have to see “where” does $\sin$ of some angle equals $\frac{1}{2}$.

And that would be $\frac{\pi}{6}$. So the correct answer is option b.

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It will help you all the time to know the values of trigonometry functions at some angles (for instance, at $0$, $\frac{\pi}{3}$, $\frac{\pi}{4}$, $\frac{\pi}{6}$...)

Answer 2

There's a sort of silly way to keep the sines of common angles in your head. The common angles are:

$$0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}.$$

The sine of each of these, in order is:

$$\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}.$$

The cosines are the reverse order, and then you have all the trig functions for these angles.

(But yes, I think it makes more sense to just know the two special triangles involved.)