Trigonometric Relations

For a trigonometric angle \(\alpha\), the trigonometric values \(\sin \alpha\), \(\cos \alpha\), \(\tan \alpha\), and \(\cot \alpha\) are related to each other by the following identities:

$\sin^2 \alpha + \cos^2 \alpha = 1$

$1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha}$ with \(\cos \alpha \ne 0\).

$1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha}$ with \(\sin \alpha \ne 0\).

$\tan \alpha \cdot \cot \alpha = 1$ with \(\cos \alpha \ne 0\) and \(\sin \alpha \ne 0\).

Example 1:

Given \(\sin \alpha = \frac{1}{3}\) and \(\frac{\pi}{2} < \alpha < \pi\). Calculate the other trigonometric values of \(\alpha\).

We have:

$\sin^2 \alpha + \cos^2 \alpha = 1 \Rightarrow \cos^2 \alpha = 1 – \sin^2 \alpha = 1 – \left(\frac{1}{3}\right)^2 = \frac{8}{9}$

$\Rightarrow \cos \alpha = \frac{2\sqrt{2}}{3}$ or \(\cos \alpha = -\frac{2\sqrt{2}}{3}\).

Since \(\frac{\pi}{2} < \alpha < \pi\) (the representation point of \(\alpha\) lies in the second quadrant), \(\cos \alpha < 0\), so \(\cos \alpha = -\frac{2\sqrt{2}}{3}\).

$\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{1}{3}}{-\frac{2\sqrt{2}}{3}} = -\frac{\sqrt{2}}{4}$

$\cot \alpha = \frac{\cos \alpha}{\sin \alpha} = \frac{-\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = -2\sqrt{2}$

(Alternatively, \(\tan \alpha \cdot \cot \alpha = 1 \Rightarrow \cot \alpha = \frac{1}{\tan \alpha} = -2\sqrt{2}).

Example 2:

Given \(\cot \alpha = \sqrt{2}\) and \(-\pi < \alpha < -\frac{\pi}{2}\). Calculate the other trigonometric values of \(\alpha\).

We have:

$1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha}$

$\Rightarrow \sin^2 \alpha = \frac{1}{1 + \cot^2 \alpha} = \frac{1}{1 + (\sqrt{2})^2} = \frac{1}{3}$

$\Rightarrow \sin \alpha = \frac{\sqrt{3}}{3}$ or \(\sin \alpha = -\frac{\sqrt{3}}{3}\).

Since \(-\pi < \alpha < -\frac{\pi}{2}\), \(\sin \alpha < 0\), so \(\sin \alpha = -\frac{\sqrt{3}}{3}\).

$\cot \alpha = \frac{\cos \alpha}{\sin \alpha}$

$\Rightarrow \cos \alpha = \sin \alpha \cdot \cot \alpha = -\frac{\sqrt{3}}{3} \cdot \sqrt{2} = -\frac{\sqrt{6}}{3}.$

$\tan \alpha = \frac{1}{\cot \alpha} = \frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$