# What is the reference angle for θ = 210◦ ? Using that reference angle, determine the vaules for the six trigonometric functions of 210◦.

Mar 1, 2018

Reference angle is ${30}^{\circ}$, angle is in QIII

$\sin {210}^{\circ} = - \frac{1}{2}$, $\csc {210}^{\circ} = - 2$

$\cos {210}^{\circ} = - \frac{\sqrt{3}}{2}$, $\sec {210}^{\circ} = - \frac{2 \sqrt{3}}{3}$

$\tan {210}^{\circ} = \frac{\sqrt{3}}{3}$, $\cot {210}^{\circ} = \sqrt{3}$

#### Explanation:

The reference angle is found by calculating the difference between $\theta$ and the x-axis. In this problem, 210 is closest to 180, so ${210}^{\circ} - {180}^{\circ} = {30}^{\circ}$. This is your reference angle. We can calculate the values of all six trig ratios using a combination of the reference angle and the quadrant in which it lies (Q3). In Q3, both $\sin \theta$ and $\cos \theta$ are negative. But $\tan \theta$ is positive. Armed with all this information, we can get the values of the six trig ratios above.