Use a sum or difference formula to find the exact value of t

Use a sum or difference formula to find the exact value of the trigonometric function without using a calculator.

Solution

Solution:

cos 19pi/12 = cos (12pi/12 + 7pi/12) reducing next

cos 19pi/12 = cos (pi + 7pi/12)

To continue you will need to know sin pi, cos pi, sin 7pi/12, cos 7pi/12.

sin pi = 0, cos pi = -1, sin 7pi/12, cos 7pi/12

You will use the sum and difference identities.

cos (a+b) = cos a cos b - sin a sin b

cos 19pi/6 = cos (pi + 7pi/12)
          = cos pi * cos 7pi/12 - sin pi*sin 7pi/12
                = -1* cos 7pi/12 - 0*sin 7pi/12

cos 19pi/6 = - cos 7pi/12

So

cos 19pi/6 = - cos 7pi/12 = - cos (3pi/12 + 4pi/12) reducing next

= - cos (pi/4 + pi/3)

cos (a+b) = cos a cos b - sin a sin b

               = - { cospi/4 * cospi/3 - sin pi/4 * sin pi/3}

               = - {(1/sqrt(2)) * (1/2) - (1/sqrt(2)) * (sqrt(3) / 2)}

              = - { (1 - sqrt(3)) / 2sqrt(2)}

              = {(sqrt(3) - 1) / 2sqrt(2)}

cos 19pi/12 = 0.2588