# Small-Angle Approximation for Cosine

The small-angle approximation for cosine is: $$\cos (x) = 1 - \frac{x^2}{2}$$ Question: How can I find a range of values of $$x$$ for which this approximation gives correct results rounded to 2 decimal places?

Thought: The error term of this $$2^{nd}$$-order Taylor approximation is $$E(x)=\frac{sin(\eta)}{6}x^3,$$ where $$\eta$$ is between $$x$$ and $$0$$. Thus, $$|E(x)|<10^{-2} \to |\sin(\eta)x^3|<6\times10^{-2}.$$ This is just my thought, but I am not sure this is the correct approach.

• Right. Since $|\eta|<|x|$ you will have $|\sin\eta|\le|\eta|\le |x|$ and you derive a sufficient condition on $x$ from here. – GReyes Feb 10 at 23:31
• Observe also that you can think of your approximation as a third order approximation (the series contains only even powers) and use the corresponding error term, which is smaller and gives you a better range for the possible values of $x$. – GReyes Feb 10 at 23:34
• @GReyes Thank you. I got the same result as that in Ross Millikan's answer. – A Slow Learner Feb 10 at 23:44

The alternating series theorem says the truncation error is smaller than the first neglected term and of the same sign. The first term you neglect is $$\frac {x^4}{4!}$$ so we want $$\frac {x^4}{4!} \lt 0.01\\x^4 \lt 0.24\\|x|\lt 0.24^{1/4}\approx 0.700$$ When you demand correct rounding to a number of places it is hard to say what the allowable error is. If you are very close to a breakpoint you may have to be very accurate. I used $$0.01$$ as the allowable error, you can use whatever value you want.
• @Just to clarify the answer, the answer should be $x \in (-0.7,0.7)$, right? Thank you. – A Slow Learner Feb 12 at 2:47
• Yes, I will put absolute value bars on the $x$ in the last line. Thanks – Ross Millikan Feb 12 at 2:56