# How do you evaluate sin(-pi/6)?

Mar 23, 2016

We should first convert to degrees using the conversion ratio $\frac{180}{\pi}$

#### Explanation:

For the time being, disregard the negative sign.

$\to \frac{\pi}{6} \times \frac{180}{\pi} = 30$ degrees

Now, referring to our special triangles, we find that 30 degrees is part of the 30-60-90; 1 sqrt(3), 2 triangle. 30 degrees is opposite the 1, adjacent the side of length $\sqrt{3}$, and the hypotenuse is 2.

The definition of sin is opposite/hypotenuse, therefore our ratio is $\frac{1}{2}$. However, we must now deal with the negative sign.

Imagine drawing your angle in standard position. If it was positive, you would go counter-clockwise. However, if it was negative you would draw it in a clock-wise direction. So, we can state that 30 and -30 degrees measure the same thing, it's just that they are to be drawn in different directions and different quadrants. By extension, the sides of the special triangle must measure the same thing, with one exception: your ratio will be $- \frac{1}{2}$, because your are in quadrant IV and therefore, sin is negative (the side opposite your angle measures -1).

Hence, $\sin \left(- \frac{\pi}{6}\right) = - \frac{1}{2}$

Hopefully this helps.