Find the exact value of cos 375 degrees.
Answer: (sqrt6 + sqrt2)/4
Is this correct?
15 deg
sin (30/2)= sqrt [(1-cos 30)/2 ]
cos 30 = sqrt 3 /2
sqrt [ (1- sqrt 3/2)/2]
sqrt [ (2 - sqrt (3) ) /4 ]
= (1/2) sqrt (2 - sqrt 3)
To verify if the provided answer is correct, we can use the unit circle and the angle sum/difference identities for cosine to find the exact value of cos 375 degrees.
The angle 375 degrees can be written as the sum of two angles: 360 degrees and 15 degrees. The value of cosine for these two angles can be determined as follows:
1. cos 360 degrees = 1 (cosine of a full revolution is always 1).
2. cos 15 degrees can be found by using trigonometric identities. We can express 15 degrees as the difference of two angles: 45 degrees and 30 degrees.
a. cos 45 degrees = sqrt(2)/2 (These values can be found in the unit circle or by using special triangles).
b. cos 30 degrees = sqrt(3)/2.
Now we can use the angle sum identity for cosine: cos(A + B) = cos A * cos B - sin A * sin B.
Applying the formula, we have:
cos 375 degrees = cos (360 degrees + 15 degrees)
= cos 360 degrees * cos 15 degrees - sin 360 degrees * sin 15 degrees
= (1) * (cos 15 degrees) - (0) * (sin 15 degrees) (since sin 360 degrees = 0)
= cos 15 degrees
= cos (45 degrees - 30 degrees)
= cos 45 degrees * cos 30 degrees + sin 45 degrees * sin 30 degrees
= (sqrt(2)/2) * (sqrt(3)/2) + (sqrt(2)/2) * (1/2)
= (sqrt(6) + sqrt(2))/4.
Therefore, the exact value of cos 375 degrees is (sqrt(6) + sqrt(2))/4.
So, the provided answer is correct.