How do you find the exact functional value csc (-15 degrees) using the cosine sum or difference identity?

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Answer 1 · 2015-08-28T00:28:50

#color(red)(csc(-15) = -sqrt2(1+sqrt3))#

#csc(-15) = -csc(15)#, since cosecant is an odd function.

#cscx = 1/sinx#

∴ #csc(-15) = -1/sin(15)#, so we can evaluate #sin(15)#.

#15 = 45-30#

∴ #sin(15) = sin(45-30)#

The sine difference identity is

#sin(A-B) = sinAcosB-cosAsinB#

∴ #sin(15) = sin(45)cos(30) – cos(45)sin(30)#

We can use the Unit Circle to evaluate these functions.

![Unit Circle]( useruploads.socratic.org
(from www.algebra.com)

#sin(15) = sqrt2/2·sqrt3/2 – sqrt2/2·1/2 = sqrt2/4(sqrt3-1) = (sqrt3-1)/(2sqrt2)#

#csc(-15) = -1/sin(15) = -(2sqrt2)/(sqrt3-1) = -(2sqrt2)/(sqrt3-1)× (sqrt3+1)/(sqrt3+1)#

#csc(-15) = -(2sqrt2(sqrt3+1))/(3-1) = -(2sqrt2(1+sqrt3))/2#

#csc(-15) = -sqrt2(1+sqrt3)#