Find the linear approximation L(x)of the function f(x)=cos(pi/(6)x) at the point x=1 and use it to estimate the value of cos(13pi/72).

Here's what I did so far:
L(x)=sqrt(3)/2-1/12pi(x-1)+0((x-1)^2)

How do I find cos(13pi/72)

f' = -pi/6 sin(pi/6 x)

f'(1) = -pi/6 (1/2) = -pi/12
f(1) = √3/2

So, the tangent line is

y-√3/2 = -pi/12 (x-1)

13pi/72 = pi/6 (13/12)

So, plug in x=13/12 into the equation of the line. Note that 13/12 is close to 1, so the approximation should be close to cos(13pi/72)

which you can get using any scientific calculator.

To find the value of cos(13pi/72) using the linear approximation, we can substitute the given value of x into the linear approximation formula L(x).

The linear approximation formula is given by: L(x) = f(a) + f'(a)(x-a)

Let's calculate step by step:

1. Find the value of f(x) = cos(pi/6 * x) at the point x=1:
f(1) = cos(pi/6 * 1) = cos(pi/6) = sqrt(3)/2

2. Calculate the derivative of f(x) = cos(pi/6 * x) with respect to x:
f'(x) = derivative of cos(u) = -sin(u) * du/dx
In this case, u = pi/6 * x
So, f'(x) = -sin(pi/6 * x) * (pi/6)

3. Calculate f'(a) by substituting a=1 into f'(x):
f'(1) = -sin(pi/6 * 1) * (pi/6)
= -sin(pi/6) * (pi/6)
= -1/2 * (pi/6)
= -pi/12

4. Substitute the values of f(a) and f'(a) into the linear approximation formula:
L(x) = f(a) + f'(a)(x-a)
L(x) = sqrt(3)/2 + (-pi/12)(x-1)

5. Substitute the given value of x = 13pi/72 into the linear approximation formula:
L(13pi/72) = sqrt(3)/2 + (-pi/12)(13pi/72 - 1)

6. Simplify the equation:
L(13pi/72) = sqrt(3)/2 + (-pi/12)(13pi/72 - 1)
= sqrt(3)/2 + (-pi/12)((13pi - 72)/72)
= sqrt(3)/2 + (-pi/12)((13pi - 72)/72)
= sqrt(3)/2 + (-pi/12)(13pi/72 - 72/72)
= sqrt(3)/2 + (-pi/12)(13pi/72 - 1)
= sqrt(3)/2 + (-pi/12)(13pi/72) + (pi/12)

7. Calculate the result:
L(13pi/72) = (sqrt(3) - 13pi/72 + pi)/2

Therefore, the linear approximation of the function f(x) at x=1 is L(x) = (sqrt(3) - 13pi/72 + pi)/2, and the estimated value of cos(13pi/72) using this linear approximation is equal to L(13pi/72).