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

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, `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 (`sqrt(3)/2`, `1/2`). Finally, the function, sin(`theta`) returns a value equal to the y-coordinate of the point, giving us an answer of `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.

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