Final answer:
To find cos(2x) when sin(x) = 1/4, use the Pythagorean identity to find cos(x) and then apply the double-angle formula cos(2x) = 1 - 2sin²(x) to get the result cos(2x) = 7/8.
Step-by-step explanation:
If sin(x) = 1/4, to find cos(2x), we can use the double-angle formula for cosine, which is cos(2x) = cos²(x) - sin²(x) or cos(2x) = 1 - 2sin²(x). As we know sin(x) is 1/4, we first need to find cos(x) which can be done using the Pythagorean identity: sin²(x) + cos²(x) = 1, thereby cos(x) = √(1 - sin²(x)). Now we can substitute sin(x) into the double-angle formula to find cos(2x).
First, find cos(x):
cos(x) = √(1 - (1/4)²) = √(1 - 1/16) = √(15/16),
Now, find cos(2x) using the second double-angle formula for convenience since we know sin(x):
cos(2x) = 1 - 2sin²(x) = 1 - 2(1/4)² = 1 - 2(1/16) = 1 - 1/8 = 7/8.
Therefore, cos(2x) = 7/8.