Determine sin105 degrees , sin7deg30'.

To calculate sin 105, we'll write 105 as the sum of 2
angles:

105 = 75 + 30

We'll
apply sine function both sides:

sin 105 = sin (75 +
30)

To calculate sin (75+30), we'll apply the
formula:

sin (a+b) = sin a*cos b + sin b*cos
a

We'll put a = 75 and b =
30

sin (75 + 30) = sin 75*cos 30 + sin 30*cos
75

We know the values for sin 30 and cos 30. We have to
calculate the values for sin 75 and cos 75.

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)

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

cos 75 = sqrt[1 -
2*(1+sqrt3)^2/16]

cos 75 = sqrt(16 - 2 - 4sqrt3 -
6)/4

cos 75 =
sqrt4(2-sqrt3)/4

cos 75 =
2sqrt(2-sqrt3)/4

cos 75 =
sqrt(2-sqrt3)/2

sin 105 = sin
75*cos 30 + sin 30*cos 75

sin
105 = sqrt6*(1+sqrt3)/8 + sqrt(2-sqrt3)/4

To
calculate cos 7deg30min, we'll write the formula for the
half-angle:

cos (a/2) = sqrt [(1+cos
a)/2]

We'll put a =
7deg30min,

2a = 2*7deg30min = 7deg30min + 7deg30min = 14deg
+ 1deg = 15 degrees

cos 7deg30min = sqrt [(1+cos
15)/2]

cos 15 = cos (30/2) = sqrt [(1+cos
30)/2]

cos (30/2) = sqrt
[(2+sqrt3)]/2

cos7deg30min = sqrt [(1 + sqrt
[(2+sqrt3)/2]/2]