Question

How do you evaluate `sin(pi/6)`?

Answer 1

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

Explanation:

Start with an equilateral triangle of side `2`. The interior angle at each vertex must be `pi/3` since `6` such angles make up a complete `2pi` circle.

Then bisect the triangle through a vertex and the middle of the opposite side, dividing it into two right angled triangles.

These will have sides of length `2`, `1` and `sqrt(2^2-1^2) = sqrt(3)`. The interior angles of each right angled triangle are `pi/3`, `pi/6` and `pi/2`, with the `pi/6` coming from the fact that we have bisected one of the `pi/3` angles.

Then:

`sin(pi/6) = "opposite"/"hypotenuse" = 1/2`

Answer 2

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

Explanation:

Use Half Angle Identity
`sin (t/2) = +- sqrt((1 - cos t)/2)`
In this case, `cos t = cos (pi/3) = 1/2` -->
`sin (pi/6) = sqrt((1 - 1/2)/2) = sqrt(1/4) = 1/2`