Question

How do you find the exact functional value csc 285º using the cosine sum or difference identity?

Answer 1

Find sin 285

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

Explanation:

`csc 285 = 1/sin 285`. Find sin 185
`sin 285 = sin (105 + 180) = - sin 105 = - sin (15 + 90) = - cos 15`
Find cos 15.
Apply the trig identity: `cos 2a = 2cos^2 a - 1`
`cos 30 = sqrt3/2 = 2cos^2 15 - 1`
`2cos^2 15 = (2 + sqrt3)/2`
`cos^2 15 = (2 + sqrt3)/4`
`cos 15 = +- sqrt(2 + sqrt3)/2`
Since the arc 285 is located in Quadrant IV, its sine is negative, then,
`sin 285 = - sqrt(2 + sqrt3)/2`
Check by calculator.
sin 285 = - 0.97
`- sqrt(2 + sqrt3)/2 = - 1.93/2 = -0.97.` OK