SOLUTION: Evaluate {{{(sin 7.5)}}} degrees exactly.

Algebra ->  Trigonometry-basics -> SOLUTION: Evaluate {{{(sin 7.5)}}} degrees exactly.       Log On


   

Question 549872: Evaluate %28sin+7.5%29 degrees exactly.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
7.5° is half of 15°, so if we knew cos(15°), we could use
the half-angle formula:

sin(theta%2F2) = %22%22+%2B-+sqrt%28%281-cos%28theta%29%29%2F2%29

sin(7.5°) = sin(%2215%B0%22%2F2) = sqrt%28%281-cos%28%2215%B0%22%29%29%2F2%29

Note we only need the positive sign since all the angles we
will be involved with are in quadrant I.

We can find cos(15°) because 15° = 45°-30°, and we know all the
trig ratios for both 45° and 30°.  We will use the identity:

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

cos(15°) = cos(45°-30°) = cos(45°)cos(30°) + sin(45°)sin(30°) =

sqrt%282%29%2F2·sqrt%283%29%2F2 + sqrt%282%29%2F2·1%2F2 = sqrt%286%29%2F4 + sqrt%282%29%2F4 = %28sqrt%286%29%2Bsqrt%282%29%29%2F4

Now we go back to the formula:

sin(7.5°) = sin(%2215%B0%22%2F2) = sqrt%28%281-cos%28%2215%B0%22%29%29%2F2%29 = sqrt%28%281-%0D%0A%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29%2F2%29  = sqrt%28%28%281-%28%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29%29%2F2%29%284%2F4%29%29 =
sqrt%28%284+-+%28sqrt%286%29+%2B+sqrt%282%29%29%29%2F8%29+ = sqrt%28%284+-+sqrt%286%29+-+sqrt%282%29%29%2F8%29+ = sqrt%28%28%284+-+sqrt%286%29+-+sqrt%282%29%29%2F8%29%282%2F2%29%29%29+ = sqrt%28%282%284+-+sqrt%286%29+-+sqrt%282%29%29%29%2F16%29+ = sqrt%282%284+-+sqrt%286%29+-+sqrt%282%29%29%29%2F4+

Edwin