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).

Question

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)

Answer 1

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.