You need to find a way to express the angle #75# degrees in terms of the standard angles.
The standard angles are (in degrees):
#30# degrees
#45# degrees
#60# degrees
#90# degrees
You need to come up with an expression that equals #75#, using only addition and dividing by 2.
One possible combination is:
#60 + 30/2 = 75#
Therefore, we can solve this by hand by using:
#cos 75#
#= cos(60+30/2)#
First we can use the sum and difference identities:
#color(green)(cos (alpha + beta) = cos alpha cos beta - sin alpha sin beta)#
#cos(60+30/2)#
#= cos 60 cos (30/2) - sin 60 sin (30/2)#
Then we can use the half angle identities:
#color(green)(sin (theta/2) = sqrt((1-cos theta)/2))#
#color(green)(cos(theta/2) = sqrt((1+cos theta)/2))#
#cos 60 cos (30/2) - sin 60 sin (30/2)#
#= cos 60 sqrt((1+cos30)/2) - sin 60 sqrt((1-cos30)/2)#
Now we can evaluate the expression by hand:
#color(green)(cos 60 = 1/2)#
#color(green)(sin 60 = sqrt(3)/2)#
#color(green)(cos 30 = sqrt(3)/2)#
#cos 60 sqrt((1+cos30)/2) - sin 60 sqrt((1-cos30)/2)#
#=(1/2) sqrt((1+sqrt(3)/2)/2) - (sqrt(3)/2) sqrt((1-sqrt(3)/2)/2)#
#= (1/2) sqrt(((2+sqrt(3))/2)/2) - (sqrt(3)/2) sqrt(((2-sqrt(3))/2)/2)#
#=(1/2) sqrt((2+sqrt(3))/4) - (sqrt(3)/2) sqrt((2-sqrt(3))/4)#
#color(blue)(=1/2 sqrt((2+sqrt(3))/4) - sqrt(3)/2 sqrt((2-sqrt(3))/4))#
