Use a halfangle formula to find the exact value of cos 165de

Use a half-angle formula to find the exact value of cos 165degree.

Solution

cos(165)= ?

cos(2x) = 2*cos²x-1

cos²x = 1+cos(2x) / 2

cos²(165) = ( 1+ cos( 2*165) ) / 2

= 1+cos330 / 2

= ( 1+ cos(360-30) ) /2

= ( 1+cos 30 ) / 2

= ( 1 + ( 3^1/2 / 2)) / 2

cos²(165) = 0.933

cos165 = +- ( 0.933)^1/2

cos165 = - 0.965 [ in second quadrant cos@ is negative)