Evaluate
cos (2 arcsin (1/4))
To evaluate the expression cos(2arcsin(1/4)), we can follow these steps:
Step 1: Determine the value of arcsin(1/4):
arcsin(1/4) can be calculated using a calculator or by using the inverse sine function. In this case, arcsin(1/4) is approximately 14.48 degrees or 0.2527 radians.
Step 2: Calculate the value of cos(2 * 0.2527):
To find cos(2 * 0.2527), we need to double the value of 0.2527. This gives us 2 * 0.2527 = 0.5054.
Step 3: Find the value of cos(0.5054):
Finally, we substitute the value of 0.5054 into the cosine function. Using a calculator or the cosine function, we find that cos(0.5054) is approximately 0.8769.
Therefore, the value of cos(2arcsin(1/4)) is approximately 0.8769.
To evaluate cos(2 arcsin(1/4)), we can start by using the trigonometric identity sin^2θ + cos^2θ = 1.
Step 1: Let's first find the value of arcsin(1/4).
Since the sine function has a range of -1 to 1, we can find the value of arcsin(1/4) by looking for an angle whose sine is 1/4.
An angle with a sine of 1/4 is approximately 14.48 degrees or 0.2527 radians.
Step 2: Now, let's find cos(2 arcsin(1/4)).
Using the double-angle identity for the cosine function, cos(2θ) = 1 - 2sin^2(θ).
Substituting θ with arcsin(1/4), we have:
cos(2 arcsin(1/4)) = 1 - 2sin^2(arcsin(1/4))
Step 3: Substitute the value of arcsin(1/4) into the equation.
cos(2 arcsin(1/4)) = 1 - 2(1/4)^2
Step 4: Simplify the equation.
cos(2 arcsin(1/4)) = 1 - 2(1/16)
cos(2 arcsin(1/4)) = 1 - 1/8
cos(2 arcsin(1/4)) = 7/8
Therefore, cos(2 arcsin(1/4)) equals 7/8.