How do you find the exact value of csc ((5pi)/6)csc(5π6) using the half angle formula?

2 Answers
Jun 29, 2016

csc5pi/6=2.csc5π6=2.

Explanation:

Half-angle formula for sinsin is : sin(theta/2)=+-sqrt{(1-costheta)/2},sin(θ2)=±1cosθ2, where sign is to be taken properly.

Putting, theta=5pi/3θ=5π3, we get,
sin{(5pi/3)/2}=sin (5pi/6)=+-sqrt{(1-cos5pi/3)/2}sin{5π32}=sin(5π6)=±1cos5π32

Since, sin(5pi/6)=sin (pi-pi/6), 5pi/6sin(5π6)=sin(ππ6),5π6 lies in the II^(nd)IInd Quadrant, +ve+ve sign has to be taken

But, cos5pi/3=cos(2pi-pi/3)=cos(pi/3)=1/2.cos5π3=cos(2ππ3)=cos(π3)=12.
:. 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