# How do you find exact value of sin (pi/ 6)?

Apr 27, 2018

$\frac{1}{2}$

#### Explanation:

For trigonometry, it is imperative to memorize a tool known as the Unit Circle. This is a circle with a radius of $1$ and a center on the origin. The points on the circumference of the circle are the coordinates that you need to know.

When you see a trigonometric function such as sine (or sin($\theta$)) or cosine (or cos($\theta$)), it refers the point on the circumference of the circle that intersects the line coming from the origin at a given angle ($\theta$) counter-clockwise from the axis between Quadrant I and Quadrant IV of the coordinate plane.

In this case, $\frac{\pi}{6}$ refers to the angle in radians, an alternate unit of measurement for angles ($\pi$ rad = 180°). The point on the unit circle that is intersected by this line is ($\frac{\sqrt{3}}{2}$, $\frac{1}{2}$). Finally, the function, sin($\theta$) returns a value equal to the y-coordinate of the point, giving us an answer of $\frac{1}{2}$.

In the future, you should memorize all the major points on the unit circle along with their reference angles and you'll be able to find these answers quickly.

Apr 28, 2018

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

#### Explanation:

The fastest way is to look at the trig table, titled "Trig Functions of Special Arcs".
This table gives --> $\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$.
Second method.
Use trig identity: sin (a - b) = sin a.cos b - sin b.cos a
$\sin \left(\frac{\pi}{6}\right) = \sin \left(\frac{\pi}{2} - \frac{\pi}{3}\right)$=
$= \sin \left(\frac{\pi}{2}\right) . \cos \left(\frac{\pi}{3}\right) - \sin \left(\frac{\pi}{3}\right) . \cos \left(\frac{\pi}{2}\right)$
Reminder. $\cos \left(\frac{\pi}{2}\right) = 0$, $\sin \left(\frac{\pi}{2}\right) = 1$, and $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$
Finally,
$\sin \left(\frac{\pi}{6}\right) = \left(1\right) \left(\frac{1}{2}\right) - 0 = \frac{1}{2}$