Use sum and difference identitites to find the exact value of co 15 degrees
oops that's (√3+1) / 2√2
Cos(15)=cos(45-30)=cos45cos30-sin45sin30
you should have committed all those to memory.
15 = 45-30
cos15 = cos45cos30 + sin45sin30
= 1/√2 * √3/2 + 1/√2 * 1/2
= (√3-1) / 2√2
To find the exact value of cos 15 degrees using the sum and difference identities, we need to express 15 degrees as a combination of angles whose cosine values are known.
The sum and difference identities for cosine are as follows:
1. cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
2. cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
We can use the identity cos(A - B) to find the exact value of cos 15 degrees by expressing 15 degrees as the difference of two angles whose cosine values are known.
Let's consider the angles 45 degrees and 30 degrees:
cos(45 - 30) = cos(45)cos(30) + sin(45)sin(30)
We know the exact values of cos 45 degrees and sin 45 degrees:
cos(45) = 1/sqrt(2)
sin(45) = 1/sqrt(2)
We also know the exact value of cos 30 degrees:
cos(30) = sqrt(3)/2
Plugging in these values, we have:
cos(45 - 30) = (1/sqrt(2))(sqrt(3)/2) + (1/sqrt(2))(1/2)
Simplifying further, we get:
cos(45 - 30) = sqrt(3)/2sqrt(2) + 1/2sqrt(2)
cos(45 - 30) = (sqrt(3) + 1)/2sqrt(2)
Therefore, cos 15 degrees is equal to (sqrt(3) + 1)/2sqrt(2).