How do you solve the following math problem: sin5° * sin40° + 1/2sin235°?
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sin(a) * sin(b) =>
(1/2) * (cos(a)cos(b) + sin(a)sin(b) - (cos(a)cos(b) - sin(a)sin(b))) =>
(1/2) * (cos(a - b) - cos(a + b))
sin(5) * sin(40) =>
(1/2) * (cos(40 - 5) - cos(40 + 5)) =>
(1/2) * (cos(35) - cos(45)) =>
(1/2) * cos(35) - (1/2) * cos(45)
(1/2) * cos(35) - (1/2) * cos(45) + (1/2) * sin(235) =>
(-1/2) * cos(45) + (1/2) * (sin(235) + cos(35))
sin(235) =>
sin(270 - 35) =>
sin(270)cos(35) - sin(35) * cos(270) =>
-1 * cos(35) - sin(35) * 0 =>
-cos(35)
(-1/2) * cos(45) + (1/2) * (-cos(35) + cos(35)) =>
(-1/2) * cos(45) + (1/2) * 0 =>
(-1/2) * cos(45)
Now, if you know your unit circle, then you know that cos(45) = sqrt(2)/2
(-1/2) * (sqrt(2)/2) =>
-sqrt(2) / 4