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

Question

Abigail Armstrong · Accepted Answer

#csc5pi/6=2.# 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.#

Madelyn Adams · Answer

= 2

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#

Nolan Alexander · Answer

undefined The half-angle formula for the cosecant function is:

$$ \csc\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{2}{\sin \theta - \sqrt{3}}} $$

Given $ \theta = \frac{5\pi}{6} $, we can find $ \sin \theta $ and substitute it into the formula.

$$ \sin \frac{5\pi}{6} = \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$

Substitute $ \sin \theta = \frac{\sqrt{3}}{2} $ into the half-angle formula:

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \sqrt{\frac{2}{\frac{\sqrt{3}}{2} - \sqrt{3}}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \sqrt{\frac{2}{\frac{\sqrt{3} - 2\sqrt{3}}{2}}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \sqrt{\frac{2}{-\frac{\sqrt{3}}{2}}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \sqrt{\frac{4}{-\sqrt{3}}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \sqrt{\frac{4}{-\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \sqrt{\frac{4\sqrt{3}}{-3}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \frac{2\sqrt{3}}{\sqrt{-3}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \frac{2\sqrt{3}}{i\sqrt{3}} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm \frac{2}{i} $$

$$ \csc\left(\frac{5\pi}{12}\right) = \pm 2i $$