The linear approximation is

The appropriated value of

is 0.974.
How to linearize a function.
The formula for linear approximation, also known as linearization, is given by:
L(x) = f(a) + f'(a) * (x - a)
L(x) is the linear approximation function.
f(x) is the original function.
a is the point around which you are linearizing.
f'(a) is the derivative of f(x) evaluated at a.
x is the variable.
Given f(x) = cos2x at π/6
when x = π/6
f(π/6) = cos(2*π/6)
= cos(π/3) or cos60⁰
= 1/2
f'(x) = derivative of f(x)
f'(x) = -2sin2x
When x = π/6
f'(π/6) = -2sin(2*π/6)
= -2sin(π/3) or sin60⁰
= -2*√3/2
= -√3
L(x) = f(a) + f'(a) * (x - a)
= 1/2 + (-√3)(x - π/6)
= 1/2 - √3x + (√3 π)/6
= -√3x + √3π/6 + 1/2
To approximate cos(1/2).
compare the functions
cos(2x) and cos(1/2)
2x = 1/2
4x = 1
x = 1/4
Substitute x = 1/4 into L(x) to find cos


= 0.974
The approximated value of

is 0.974