Question

Express as a single fraction cos 105

Answer 1

Use the unit circle when you need exact values.

105° is not one of the values given on the unit circle, but 45° and 60° are. Since 105° = 45° + 60°, you can use the "angle addition formula"

cos(α+β) = cos(α)×cos(β) - sin(α)×sin(β)

cos(45°) = sin(45°) = √2/2
cos(60°) = 1/2
sin(60°) = √3/2

cos(105°) = cos(45°)×cos(60°) - sin(45°)×sin(60°)
= (√2/2)(1/2) - (√2/2)(√3/2)
= √2/4 - √2√3/4
= (√2 - √6)/4

Answer 2

cos105=cos(90+15)
=> cos105=cos90cos15-sin90sin15
=>cos105=-sin15