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

2 Answers
Jun 29, 2016

#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.#

Jun 30, 2016

= 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#