# What are the six trig function values of pi/3?

Nov 26, 2015

The $6$ trigonometric values of $\frac{\pi}{3}$ are:

$\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
$\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$
$\tan \left(\frac{\pi}{3}\right) = \sqrt{3}$
$\csc \left(\frac{\pi}{3}\right) = \frac{2 \sqrt{3}}{3}$
$\sec \left(\frac{\pi}{3}\right) = 2$
$\cot \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{3}$

#### Explanation:

The $6$ trigonometric ratios are:

$1. \sin \theta$
$2. \cos \theta$
$3. \tan \theta$
$4. \csc \theta$
$5. \sec \theta$
$6. \cot \theta$

Using the ratios, we can determine their values when $\theta$ is $\frac{\pi}{3}$:

Recall that $\pi$ radians is ${180}^{\circ}$.

$1. \sin \theta$

$= \sin \left(\frac{\pi}{3}\right)$

$= \sin \left({180}^{\circ} / 3\right)$

$= \sin \left({60}^{\circ}\right)$

$= \frac{\sqrt{3}}{2}$

$2. \cos \theta$

$= \cos \left(\frac{\pi}{3}\right)$

$= \cos \left({180}^{\circ} / 3\right)$

$= \cos \left({60}^{\circ}\right)$

$= \frac{1}{2}$

$3. \tan \theta$

$= \tan \left(\frac{\pi}{3}\right)$

$= \tan \left({180}^{\circ} / 3\right)$

$= \tan \left({60}^{\circ}\right)$

$= \frac{\sqrt{3}}{1}$

$= \sqrt{3}$

$4. \csc \theta$

$= \frac{1}{\sin} \theta$

$= \frac{1}{\sin} \left(\frac{\pi}{3}\right)$

$= \frac{1}{\sin} \left({180}^{\circ} / 3\right)$

$= \frac{1}{\sin} \left({60}^{\circ}\right)$

$= \frac{1}{\frac{\sqrt{3}}{2}}$

$= \frac{2}{\sqrt{3}}$

$= \frac{2 \sqrt{3}}{3}$

$5. \sec \theta$

$= \frac{1}{\cos} \theta$

$= \frac{1}{\cos} \left(\frac{\pi}{3}\right)$

$= \frac{1}{\cos} \left({180}^{\circ} / 3\right)$

$= \frac{1}{\cos} \left({60}^{\circ}\right)$

$= \frac{1}{\frac{1}{2}}$

$= 2$

$6. \cot \theta$

$= \frac{1}{\tan} \theta$

$= \frac{1}{\tan} \left(\frac{\pi}{3}\right)$

$= \frac{1}{\tan} \left({180}^{\circ} / 3\right)$

$= \frac{1}{\tan} \left({60}^{\circ}\right)$

$= \frac{1}{\frac{\sqrt{3}}{1}}$

$= \frac{1}{\sqrt{3}}$

$= \frac{\sqrt{3}}{3}$