Assume we know the following:
cos(22.5 degrees)= sqrt(2+sqrt2)/2
sin(22.5 degrees)=sqrt(2-sqrt2)/2
cos(18 degrees)=sqrt((sqrt5+5)/8)
sin(18 degrees)=(sqrt5-1)/4
What is:
Sin(82.5 degrees)
cos(82.5 degrees)
sin(12 degrees)
cos(12 degrees)
using the same addition formulas as before, note that
82.5 = 60 + 22.5
12 = 30-18
To find the values of trigonometric functions at angles that are not directly given, we can use the trigonometric identities and properties to calculate them.
1. Sin(82.5 degrees):
We know that sin(82.5 degrees) = sin(90 degrees - 7.5 degrees). By using the angle sum identity for sine, sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B), we can express sin(82.5 degrees) in terms of the values we are given:
sin(82.5 degrees) = sin(90 degrees - 7.5 degrees)
= sin(90 degrees) * cos(7.5 degrees) - cos(90 degrees) * sin(7.5 degrees)
= 1 * cos(7.5 degrees) - 0 * sin(7.5 degrees)
= cos(7.5 degrees)
2. Cos(82.5 degrees):
Similar to sin(82.5 degrees), we can use the angle sum identity for cosine, cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B), to express cos(82.5 degrees) in terms of the known values:
cos(82.5 degrees) = cos(90 degrees - 7.5 degrees)
= cos(90 degrees) * cos(7.5 degrees) + sin(90 degrees) * sin(7.5 degrees)
= 0 * cos(7.5 degrees) + 1 * sin(7.5 degrees)
= sin(7.5 degrees)
3. Sin(12 degrees):
We don't have direct values for sin(12 degrees) in the given information. However, we can use the double angle formula to find it. The double angle formula for sine is sin(2A) = 2sin(A)cos(A). By setting A = 6 degrees, we can solve for sin(12 degrees):
sin(12 degrees) = 2sin(6 degrees)cos(6 degrees)
= 2[sin(90 degrees - 12 degrees)/2]cos(6 degrees)
= 2[sin(78 degrees)/2]cos(6 degrees)
= sin(78 degrees)cos(6 degrees)
4. Cos(12 degrees):
Similar to sin(12 degrees), we can use the double angle formula for cosine, cos(2A) = cos^2(A) - sin^2(A), to find the value of cos(12 degrees) by setting A = 6 degrees:
cos(12 degrees) = cos(2 * 6 degrees)
= cos^2(6 degrees) - sin^2(6 degrees)
To compute the values of sin(7.5 degrees), sin(78 degrees), cos(7.5 degrees), cos(6 degrees), we need additional calculations or use specialized tables, calculators, or software that can handle trigonometric functions with higher precision.