Question

How do you find the exact functional value csc(-255 degrees) using the cosine sum or difference identity?

Answer 1

Find `csc (-255)`

Ans: `2/sqrt(2 + sqrt3)`

Explanation:

`csc (-244) = 1/sin (-255).` Find sin (-255).
sin (-255) = sin (105 - 360) = sin 105 = sin (15 + 90) = cos 15
Apply the trig identity: `cos 2a = 2cos^2 a - 1`
`cos 30 = sqrt3/2 = 2cos^2 15 - 1`
`2cos^2 15 = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 15 = (2 + sqrt3)/4` --> `cos 15 = +- sqrt(2 + sqrt3)/2`
Since cos 15 is positive, then
`sin (-255) = cos 15 = sqrt(2 + sqrt3)/2`
`csc (-255) = 1/sin (-255) = 2/sqrt(2 + sqrt3)`
Check by calculator.
sin (-255) = 0.97
`sqrt(2 + sqrt3)/2 = sqrt(3.732) = 1.93/2 = 0.97.` OK