Answer:
[tex]\frac{1}{4}[/tex] ( [tex]\sqrt{6}[/tex] - [tex]\sqrt{2}[/tex] )
Step-by-step explanation:
Using the sine addition formula
sin(a + b) = sinacosb + cosasinb
and the exact values
sin45° = cos45° = [tex]\frac{\sqrt{2} }{2}[/tex] , sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , cos60° = [tex]\frac{1}{2}[/tex]
Given
sin165°
Note that 165° = (120 + 45)°, thus
= sin(120 + 45)°
= sin120°cos45° + cos120°sin45°
= sin60°cos45° - cos60°sin45°
= ( [tex]\frac{\sqrt{3} }{2}[/tex] × [tex]\frac{\sqrt{2} }{2}[/tex] ) - ( [tex]\frac{1}{2}[/tex] × [tex]\frac{\sqrt{2} }{2}[/tex] )
= [tex]\frac{\sqrt{6} }{4}[/tex] - [tex]\frac{\sqrt{2} }{4}[/tex]
= [tex]\frac{1}{4}[/tex] ( [tex]\sqrt{6}[/tex] - [tex]\sqrt{2}[/tex] )
