Question

How do you find the exact values of sin^3(pi/6) using the half angle formula?

Answer 1

`sin^3(pi/6) = 1/8`

Explanation:

Using the half-angle formula (as requested)
`color(white)("XXXX")` `sin(pi/6)= sqrt((1-cos(pi/3))/2`

`color(white)("XXXX")` `color(white)("XXXX")` `= sqrt((1-1/2)/2) = sqrt(1/4) = 1/2`

So `sin^3(pi/6) = (1/2)^3 = 1/8`

...although I don't understand why you would know `cos(pi/3)` and not `sin(pi/6)`; `pi/3` and `pi/6` are both standard angles

Answer 2

Find `sin^3 (pi/6)`

Ans: 1/8

Explanation:

Use trig identity:`sin^2 x = (1 - cos 2x)/2`

Trig table --> cos (2pi/6) = cos (pi/3) = 1/2

`sin^2 (pi/6) = (1 - 1/2)/2 = (2 - 1)/4 = 1/4`
`sin (pi/6) = sqrt(1/4) = 1/2`

Finally: `sin (pi/6).sin^2 (pi/6) = sin^3 (pi/6) = 1/2(1/4) = 1/8`