How do you solve cos(2arcsin 1/4) using inverse trig. functions??!!
PLEASE HELP ME!
To solve the expression cos(2arcsin(1/4)) using inverse trigonometric functions, we first need to transform the expression using the double-angle identity for cosine.
The double-angle identity for cosine states that cos(2θ) = 1 - 2sin²(θ).
In this case, θ represents arcsin(1/4). To find the value of θ, we need to use the inverse trigonometric function arcsin.
Now, let's start by finding θ:
arcsin(1/4) = θ
To solve for θ, we will use arcsin on both sides:
sin(arcsin(1/4)) = sin(θ)
Using the property of inverse trigonometric functions where sin(arcsin(x)) = x, we have:
1/4 = sin(θ)
Now we can substitute the value of sin(θ) into the double-angle identity for cosine:
cos(2arcsin(1/4)) = 1 - 2sin²(arcsin(1/4))
Substituting sin(θ) = 1/4:
cos(2arcsin(1/4)) = 1 - 2(1/4)²
Simplifying:
cos(2arcsin(1/4)) = 1 - 2(1/16)
cos(2arcsin(1/4)) = 1 - 2/16
cos(2arcsin(1/4)) = 1 - 1/8
cos(2arcsin(1/4)) = 7/8
Therefore, cos(2arcsin(1/4)) = 7/8.