How do you evaluate #csc((5pi)/3)#?

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Answer 1 · 2016-08-01T17:22:40

Let's first convert to degrees, which I personally find easier to work with. Use the conversion factor #180/pi#.

#(5pi)/3 xx 180/pi#

#=300˚#

Recall that #csctheta = 1/sintheta#, so #csc300˚ = 1/(sin300˚)#.

The reference angle for #300˚# is #60˚ -> sin60˚ = sqrt(3)/2#. Since #300˚# is in quadrant IV, #sin# is negative, so #sin300˚ = -sqrt(3)/2#.

However, since #csc300˚ = 1/(sin300˚)#, #csc300˚ = 1/(sqrt(3)/2) = 2/sqrt(3) = (2sqrt(3))/3#.

Hopefully this helps!