Principal values
Prerequisite knowledge
Principal values of $ \sin^{-1} x $:
$ -90^\circ \le \sin^{-1} x \le 90^\circ $
Principal values of $ \cos^{-1} x $:
$ 0^\circ \le \cos^{-1} x \le 180^\circ $
Principal values of $ \tan^{-1} x $:
$ -90^\circ < \tan^{-1} x < 90^\circ $
Special angles:
| $30^\circ \text{ or } {\pi \over 6}$ | $45^\circ \text{ or } {\pi \over 4}$ | $60^\circ \text{ or } {\pi \over 3}$ | |
|---|---|---|---|
| $\sin \theta$ | $ \phantom{.} {1 \over 2} $ | $ \phantom{.} {1 \over \sqrt{2}} $ | $ \phantom{.} {\sqrt{3} \over 2} $ |
| $\cos \theta$ | $ \phantom{.} {\sqrt{3} \over 2} $ | $ \phantom{.} {1 \over \sqrt{2}} $ | $ \phantom{.} {1 \over 2} $ |
| $\tan \theta$ | $ \phantom{.} {1 \over \sqrt{3}} $ | $ \phantom{.} 1 $ | $ \phantom{.} \sqrt{3} $ |
Calculating principal values
For $ \sin^{-1} x $:
$$ \boxed{ -90^\circ \le \sin^{-1} x \le 90^\circ } $$
If $x$ is positive, then $ \sin^{-1} x $ lies in the first quadrant such that $ 0^\circ \le \sin^{-1} x \le 90^\circ $.
If $x$ is negative, then $ \sin^{-1} x $ lies in the fourth quadrant such that $ -90^\circ \le \sin^{-1} x \le 0^\circ $.
For $\cos^{-1} x$:
$$ \boxed{ 0^\circ \le \cos^{-1} x \le 180^\circ } $$
If $x$ is positive, then $ \cos^{-1} x $ lies in the first quadrant such that $ 0^\circ \le \cos^{-1} x \le 90^\circ $.
If $x$ is negative, then $ \cos^{-1} x $ lies in the second quadrant such that $ 90^\circ \le \cos^{-1} x \le 180^\circ $.
For $\tan^{-1} x$:
$$ \boxed{ -90^\circ < \tan^{-1} x < 90^\circ } $$
If $x$ is positive, then $ \tan^{-1} x $ lies in the first quadrant such that $ 0^\circ \le \tan^{-1} x < 90^\circ $.
If $x$ is negative, then $ \tan^{-1} x $ lies in the fourth quadrant such that $ -90^\circ < \tan^{-1} x \le 0^\circ $.
Questions
1. Find, without the use of a calculator, the principal value of sin⁻¹ (-0.5) in radians.
Answer: $ -{\pi \over 6} $
2. Without using a calculator, find, in radians, the principal value of $ \sin^{-1} \left( \cos {5 \pi \over 6} \right) $.
Answer: $ -{\pi \over 3} $
Past year O level questions
| Year & paper | Comments |
|---|---|
| 2016 P1 Question 3a | State the range of principal values (quite easy) |