Linear approximation of $\cos\big(\frac{\pi}{5}+0.07\big)-\cos\big(\frac{\pi}{5}\big)$

Question

Find the approximate value of $$\cos\bigg(\frac{\pi}{5}+0.07\bigg)-\cos\bigg(\frac{\pi}{5}\bigg)$$ using linear approximation.

My attempt:

The tangent line approximation of $f(x)=\cos(x)$ at $\displaystyle x=\frac{\pi}{5}$ is

$$f(x)\approx f\bigg(\frac{\pi}{5}\bigg)+\bigg(x-\frac{\pi}{5}\bigg)f'\bigg(\frac{\pi}{5}\bigg).$$

Putting $x=\frac{\pi}{5}+0.07$, we get

$$f(x)-f\bigg(\frac{\pi}{5}\bigg)\approx -\sin\bigg(\frac{\pi}{5}\bigg)(0.07)=0.0411$$

Is my solution is right? If not, then how do I solve it?

Answer 1

You're missing the negative sign, otherwise it's correct.

In fact we have $\cos(\frac{\pi}{5}+0.07)-\cos(\frac{\pi}{5})\approx -0.0430926$ which is not too far away from the linear approximation.

Answer 2

Here's another way to approach it. For very a small angle $\theta \approx 0$, we have $\cos\theta \approx 1$ and $\sin\theta \approx \theta$. You can verify that these are, in fact, the linear approximations to $f(\theta)=\cos\theta$ and $f(\theta)=\sin\theta$ near $\theta=0$.

Then, using the angle sum identity for cosine, i.e., $\cos(\alpha+\beta) = \cos\alpha\cos\beta-\sin\alpha\sin\beta$,

$$ \begin{align} \cos\left(\frac\pi5+0.07\right)&=\cos\frac\pi5\cos0.07-\sin\frac\pi5\sin0.07\\ &\approx\cos\frac\pi5-0.07\sin\frac\pi5, \end{align} $$

giving you the same result.