Expert: Steve Holleran - 2/26/2008

Question
Hi again!  My second question involves finding the exact value of sin105cos15, but I can't use a calculator, which I am used to using, so I am stuck.  Thanks again!

Answer
Hi Rich,

To do these types of problems, you want to make use of the addition/subtraction formulas for sine and cosine:

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

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

Now, the trick here is to use angles that add to 105 and subtract to give 15, and you want to think of the special angles that you can find values by diagramming reference triangles.  Here, you want to use 60 degrees and 45 degrees:

sin105 = sin(60 + 45) = sin 60 cos 45 + cos 60 sin 45

                     = sqrt(3)/2 * sqrt(2)/2 + 1/2 * sqrt(2)/2

                     =sqrt(6) / 4 + sqrt(2) / 4

                     = [sqrt(6) + sqrt(2)] / 4

cos15 = cos(60 - 45) = cos 60 cos 45 + sin 60 sin45

                    = 1/2 * sqrt(2)/2 + sqrt(3)/2 * sqrt(2)/2

                    = sqrt(2) / 4 + sqrt(6) / 4

                    = [sqrt(2) + sqrt(6)] / 4

Then the product would be :

             [6 + 2sqrt(12) + 2] / 16

=             [8 + 4sqrt(3)]/ 16

=               [2 + sqrt(3)] / 4


Hope this helps out.
Steve