Question

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

Answer 1

`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, `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.

Answer 2

`sin (pi/6) = 1/2`

Explanation:

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`