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

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?

• Yea this is correct :) Aug 12, 2020 at 18:23

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.

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.

• It ends up being the same as linear approximations, but it's not exactly showing an understanding of linear approximations. Aug 13, 2020 at 3:02
• @Acccumulation Presumably the OP did the question as it was intended, so my answer is more of a side comment just to show another way. It uses linear approximations as well; I added a sentence to make that clear. Aug 13, 2020 at 3:25
• I think this answer provides more insight than the mechanical calculation with derivatives. Unlike the latter, it connects the approximation directly to the Maclaurin series for the trigonometric functions, which --depending on the text in use-- may be the definitions used. Aug 13, 2020 at 4:29