To calculate sin 75, we'll write 75 as the sum of 2
angles:
75 = 30 + 45
We'll
apply sine function both sides:
sin 75 = sin
(30+45)
To calculate sin (30+45), we'll apply the
formula:
sin (a+b) = sin a*cos b + sin b*cos
a
We'll put a = 30 and b =
45
sin (30+45) = sin 30*cos 45 + sin 45*cos
30
We'll substitute sin 30; sin 45; cos 30; cos 45 by their
values:
sin 30 = 1/2
cos 30 =
sqrt3/2
sin 45 = cos 45 =
sqrt2/2
sin (30+45) = (1/2)*(sqrt2/2) +
(sqrt2/2)*(sqrt3/2)
We'll factorize by
(sqrt2/2):
sin (30+45) =
(sqrt2/2)[(1+sqrt3)/2]
sin (30+45) =
sqrt2*(1+sqrt3)/4
sin 75 =
sqrt2*(1+sqrt3)/4
To calculate
cos22deg30min, we'll write the formula for teh
half-angle:
cos (a/2) = sqrt [(1+cos
a)/2]
We'll put a =
22deg30min
2a = 2*22deg30min = 22deg30min + 22deg30min =
44deg + 1deg = 45 degrees
cos22deg30min = sqrt [(1+cos
45)/2]
cos22deg30min = sqrt [(1 +
sqrt2/2)/2]
cos22deg30min =
sqrt(2+sqrt2)/2
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