find cos x/2, sin x = 1/4, 360 < x < 450, use half-angle identity
To find cos(x/2) using the given information and the half-angle identity, follow these steps:
1. Recall the half-angle identity for cosine: cos(x/2) = ±√((1 + cos(x)) / 2)
2. Start by finding cos(x) using the given value for sin(x). To do this, use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Given: sin(x) = 1/4
Plug this value into the Pythagorean identity:
(1/4)^2 + cos^2(x) = 1
1/16 + cos^2(x) = 1
cos^2(x) = 1 - 1/16
cos^2(x) = 16/16 - 1/16
cos^2(x) = 15/16
Now, take the square root of both sides to solve for cos(x):
cos(x) = ±√(15/16)
Note that since x is between 360 and 450 degrees, the cosine value should be negative.
3. Plug the value of cos(x) into the half-angle identity for cosine:
cos(x/2) = ±√((1 + cos(x)) / 2)
cos(x/2) = ±√((1 + (√(15/16))) / 2)
cos(x/2) = ±√((1 + (√15/4)) / 2)
cos(x/2) = ±√((1 + (√15))/2√4)
cos(x/2) = ±√((1 + (√15))/4)
cos(x/2) = ±√((1 + (√15))/4)
Finally, since x is between 360 and 450 degrees, cos(x/2) should be negative. Therefore:
cos(x/2) = -√((1 + (√15))/4)
Note: The square root of 15 (√15) cannot be simplified further as it is an irrational number.