Assume that we know sin(18 degrees)=(sqrt(5)-1)/4, cos(18 degrees)=(sqrt(sqrt(5)+5))/8, find sin(42 degrees) exactly, without using a calculator. Also find cos(42 degrees).
To find sin(42 degrees) and cos(42 degrees) exactly without using a calculator, we can use the trigonometric identities and the values given for sin(18 degrees) and cos(18 degrees).
First, we can use the sum of angles identity for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Let's represent 42 degrees as the sum of 18 degrees and 24 degrees: 42 degrees = 18 degrees + 24 degrees.
sin(42 degrees) = sin(18 degrees + 24 degrees)
Using the sum of angles identity, we have:
sin(42 degrees) = sin(18 degrees)cos(24 degrees) + cos(18 degrees)sin(24 degrees)
Now, we substitute the given values:
sin(42 degrees) = (sqrt(5) - 1)/4 * cos(24 degrees) + (sqrt(sqrt(5) + 5))/8 * sin(24 degrees)
For cos(42 degrees), we can use the cos(A + B) identity: cos(A + B) = cos(A)cos(B) - sin(A)sin(B):
cos(42 degrees) = cos(18 degrees + 24 degrees)
Using the cos(A + B) identity, we have:
cos(42 degrees) = cos(18 degrees)cos(24 degrees) - sin(18 degrees)sin(24 degrees)
Now, let's substitute the given values:
cos(42 degrees) = (sqrt(sqrt(5) + 5))/8 * cos(24 degrees) - (sqrt(5) - 1)/4 * sin(24 degrees)
To find the exact values of sin(42 degrees) and cos(42 degrees), we need to know the exact values of cos(24 degrees) and sin(24 degrees). However, these values are not provided in the information given.