Question

How do I find the value of csc pi/12?

Answer 1

Find `csc (pi/12)`

Ans: `2/sqrt(2 - sqrt3)`

Explanation:

`csc (pi/12) = 1/sin (pi/12)`. Find `sin (pi/12)`
`call sin (pi/12) = sin t`
`cos 2t = cos ((2pi)/12) = cos ((pi)/6) = sqrt3/2.`
Use trig identity: `cos 2t = sqrt3/2 = 1 - 2sin^2 t`
`2sin^2 t = 1 - sqrt3/2` --> `sin^2 t = (2 - sqrt3)/4`
`sin (pi/12) = sin t = +- sqrt(2 - sqrt3)/2`
Since sin `(pi/12)` is in Quadrant I, therefor, only the positive answer is accepted.

`csc (pi/12) = 2/(sqrt(2 - sqrt3)`