We'll write 75 as a sum of 2
angles:
75 = 45 + 30
We'll
apply sine function both sides:
sin(75) =
sin(45+30)
We'll use the
identity:
sin(a+b) = sin a*cos b + sin b*cos
a
sin(45+30) = sin 45*cos 30 + sin 30*cos
45
sin(45+30) = (sqrt2/2)*(sqrt3/2) +
(1/2)*(sqrt2/2)
sin(45+30) = sqrt(2*3)/4 +
sqrt2/4
sin(45+30) = (sqrt6 +
sqrt2)/4
We notice that we get the indicated
result, therefore the identity sin(75) = (sqrt6 + sqrt2)/4 is
verified.
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