Question

Use the appropriate sum/difference formula to find the exact value of cos(105°).

do we choose one, then make 1 side equal cos of 105°? I don't understand the question.

Answer 1

105 ° = 210 ° / 2

210 ° = 180 ° + 30 °


cos ( A + B ) = cos ( A ) * cos ( B ) - sin ( A ) * sin ( B )


sin 30 ° = 1 / 2

sin 180 ° = 0

cos 30 ° = sqrt ( 3 ) / 2

cos 180 ° = - 1


cos 210 ° = cos ( 180 ° + 30 ° ) = cos 180 ° * cos 30 ° - sin 180 ° * sin 30 ° =

( - 1 ) * sqrt ( 3 ) / 2 - 0 * 1 / 2 =

- sqrt ( 3 ) / 2 - 0


cos 210 ° = - sqrt ( 3 ) / 2


cos ( theta / 2 ) = ± sqrt [ ( 1 / 2 ) * ( 1 + cos ( theta )) ]

cos 105 ° = cos ( 210 ° / 2 ) =

± sqrt [ ( 1 / 2 ) * ( 1 + ( - sqrt ( 3 ) / 2 ) ] =

± sqrt [ ( 1 / 2 ) * ( 2 / 2 + ( - sqrt ( 3 ) / 2 ) ] =

± sqrt [ ( 1 / 2 ) * ( ( 2 - sqrt ( 3 ) / 2 ) ) ] =

± sqrt [ ( ( 2 - sqrt ( 3 ) ) / 4 ] =

± ( 1 / 2 ) sqrt [ 2 - sqrt ( 3 ) ] =

± sqrt [ 2 - sqrt ( 3 ) ] / 2



The angle 105 ° is in the II quadrant

In quadrant II cosine is negative.

So

cos ( 105 ° ) = - sqrt [ 2 - sqrt ( 3 ) ] / 2