Question

Find the exact values of the sine, cosine, and tangent of the angle. 255=300-45

Answer 1

Sin 255 = sin(300 - 45) = sin 300 cos 45 - cos 300 sin 45; where sin 300 = -sin(360 - 60) = - sin 60 = -√3/2, sin 45 = cos 45 = 1/√2, cos 300 = cos(360 - 60) = cos 60 = 1/2
Therefore, sin 255 = (-√3/2)(1/√2) - (1/2)(1/√2) = -√3/2√2 - 1/2√2 = -(√3 + 1)/2√2 = -(√6 + √2)/4

cos 255 = cos(300 - 45) = cos 300 cos 45 + sin 300 sin 45 = (1/2)(1/√2) + (-√3/2)(1/√2) = (1 - √3)/2√2 = (√2 - √6)/4

tan 255 = tan(300 - 45) = (tan 300 - tan 45)/(1 + tan 300 tan 45); where tan 300 = sin 300 / cos 300 = (-√3/2) / (1/2) = -√3 and tan 45 = 1
Therefore, tan 255 = (-√3 - 1)/(1 + (-√3)) = (-√3 - 1)/(1 - √3) = √3 + 2