Using appropriate power series, calculate to accuracy
Using appropriate power series, calculate to accuracy $0.0001$: $\cos10^\circ$; $\sin1^\circ$; $\sin\frac\pi4$; $\arcsin1$; $\arctan\frac15$; $\sqrt e$.
Explanation of Method
Each value is computed by substituting the appropriate (small, in radians) argument into the relevant Maclaurin series and summing terms until the next term is smaller than the required tolerance.
- Convert degree measures to radians before substituting into $\sin$/$\cos$ series.
- Sum each series until the omitted term is below $0.0001$.
- Handle $\arcsin1$ separately, since it sits at the boundary of convergence.
$\cos10^\circ$
With $x=\frac\pi{18}\approx0.174533$: $$ \cos x = 1-\frac{x^2}2+\frac{x^4}{24}-\cdots = 1-0.015231+0.000039-\cdots \approx 0.984808. $$ Boxed: $\cos10^\circ\approx0.9848$.
$\sin1^\circ$
With $x=\frac\pi{180}\approx0.017453$: since $x^3/6\approx0.00000089<0.0001$, one term suffices: $\sin1^\circ\approx0.017453\approx0.0175$.
$\sin\frac\pi4$
With $x=\frac\pi4\approx0.785398$, $x^2\approx0.616850$: $$ x^3\approx0.484473,\ \ x^5\approx0.298847,\ \ x^7\approx0.184344,\ \ x^9\approx0.113724. $$ $$ \sin\frac\pi4 \approx x-\frac{x^3}6+\frac{x^5}{120}-\frac{x^7}{5040}+\frac{x^9}{362880} \approx 0.785398-0.080746+0.002490-0.0000366+0.0000003 \approx 0.707107. $$ Matches the exact value $\frac{\sqrt2}2\approx0.707107$. Boxed: $\sin\frac\pi4\approx0.7071$.
$\arcsin1$
The series for $\arcsin x$ converges (slowly) only up to the boundary $x=1$; rather than summing it there, it is faster to use the exact value $\arcsin1=\frac\pi2\approx1.5708$ directly.
$\arctan\frac15$
With $x=0.2$: $$ \arctan\frac15 \approx 0.2-0.002667+0.000064-0.0000018 \approx 0.197397. $$ Boxed: $\arctan\frac15\approx0.1974$.
$\sqrt e$
Using $\sqrt e=e^{1/2}=\sum\frac{(1/2)^n}{n!}$: $$ 1+0.5+0.125+0.020833+0.002604+0.000260+0.000022 \approx 1.648720. $$ Boxed: $\sqrt e\approx1.6487$.
Summary Table
| Quantity | Value |
|---|---|
| $\cos10^\circ$ | $0.9848$ |
| $\sin1^\circ$ | $0.0175$ |
| $\sin\frac\pi4$ | $0.7071$ |
| $\arcsin1$ | $\pi/2\approx1.5708$ |
| $\arctan\frac15$ | $0.1974$ |
| $\sqrt e$ | $1.6487$ |
Additional Notes
- Small-angle quantities ($\sin1^\circ$) need only one nonzero term, while quantities near the edge of a series' natural scale ($\sin\frac\pi4$, $\sqrt e$ at $x=1/2$) need several — the number of terms required tracks how large the argument is relative to $1$.
- $\arcsin1$ illustrates that a series' boundary of convergence is not always the most efficient way to evaluate a function there — using the known closed form $\pi/2$ is both faster and exact.
- This kind of term-counting exercise is the practical companion to the abstract remainder bound of Problem 281 and the explicit error estimates of Problems 279–280.