Question

How do you find the exact value of `csc ((5pi)/6)` using the half angle formula?

Answer 1

`csc5pi/6=2.`

Explanation:

Half-angle formula for `sin` is : `sin(theta/2)=+-sqrt{(1-costheta)/2},` where sign is to be taken properly.

Putting, `theta=5pi/3`, we get,
`sin{(5pi/3)/2}=sin (5pi/6)=+-sqrt{(1-cos5pi/3)/2}`

Since, `sin(5pi/6)=sin (pi-pi/6), 5pi/6` lies in the `II^(nd)` Quadrant, `+ve` sign has to be taken

But, `cos5pi/3=cos(2pi-pi/3)=cos(pi/3)=1/2.`
`:. sin5pi/6=sqrt{(1-1/2)/2}=sqrt(1/4)=1/2.`

Hence, `csc(5pi/6)=1/sin(5pi/6)=2.`

Answer 2

= 2

Explanation:

We can evaluate csc ((5pi)/6) without using half angle formula.
`csc ((5pi)/6) = 1/sin ((5pi)/6)`.
Find `sin ((5pi)/5).`
Trig table, and unit circle -->
`sin ((5pi)/6) = sin (-pi/6 + (6pi)/6) = sin (-pi/6 + pi) =`
`= sin (pi/6) = 1/2`
Therefor,
`csc ((5pi)/6) = 1/(sin) = 2`