233. The series for $\cos z$ and $\sin z$

1. Calculate $\cos i$ and $\sin i$ to two places of decimals by means of the power series for $\cos z$ and $\sin z$.

2. Prove that $|\cos z|\le \cosh |z|$ and $|\sin z|\le \sinh |z|$.

3. Prove that if $|z|<1$ then $|\cos z|<2$ and $|\sin z|<\frac{6}{5}|z|$.

4. Since $\sin 2z=2\sin z\cos z$ we have $$(2z)–\frac{(2z{)}^3}{3!}+\frac{(2z{)}^5}{5!}–\cdots =2(z–\frac{z^3}{3!}+\dots )(1–\frac{z^2}{2!}+\dots ).$$ Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that $$(\genfrac{}{}{0ex}{}{2n+1}{1})+(\genfrac{}{}{0ex}{}{2n+1}{3})+\cdots +(\genfrac{}{}{0ex}{}{2n+1}{2n+1})=2^{2n}.$$ Verify the result by means of the binomial theorem. Derive similar identities from the equations $${\cos }^{2}z+{\sin }^{2}z=1,\cos 2z=2{\cos }^{2}z–1=1–2{\sin }^{2}z.$$

5. Show that $$\exp \{(1+i)z\}=\sum _0^{\mathrm{\infty }}{2}^{\frac{1}{2}n}\exp (\frac{1}{4}n\pi i)\frac{z^n}{n!}.$$

6. Expand $\cos z\cosh z$ in powers of $z$. [We have $$\begin{array}{rl}\cos z\cosh z+i\sin z\sinh z& =\cos \{(1–i)z\}=\frac{1}{2}[\exp \{(1+i)z\}+\exp \{-(1+i)z\}]\\ & =\frac{1}{2}\sum _0^{\mathrm{\infty }}{2}^{\frac{1}{2}n}\{1+(-1{)}^n\}\exp (\frac{1}{4}n\pi i)\frac{z^n}{n!},\end{array}$$ and similarly $$\cos z\cosh z–i\sin z\sinh z=\cos (1+i)z=\frac{1}{2}\sum _0^{\mathrm{\infty }}{2}^{\frac{1}{2}n}\{1+(-1{)}^n\}\exp (-\frac{1}{4}n\pi i)\frac{z^n}{n!}.$$ Hence $$\cos z\cosh z=\frac{1}{2}\sum _0^{\mathrm{\infty }}{2}^{\frac{1}{2}n}\{1+(-1{)}^n\}\cos \frac{1}{4}n\pi \frac{z^n}{n!}=1–\frac{2^2{z}^4}{4!}+\frac{2^4{z}^8}{8!}–\dots .]$$

7. Expand $\sin z\sinh z$, $\cos z\sinh z$, and $\sin z\cosh z$ in powers of $z$.

8. Expand ${\sin }^{2}z$ and ${\sin }^{3}z$ in powers of $z$. [Use the formulae $${\sin }^{2}z=\frac{1}{2}(1–\cos 2z),{\sin }^{3}z=\frac{1}{4}(3\sin z–\sin 3z),\dots .$$ It is clear that the same method may be used to expand ${\cos }^{n}z$ and ${\sin }^{n}z$, where $n$ is any integer.]

9. Sum the series $$C=1+\frac{\cos z}{1!}+\frac{\cos 2z}{2!}+\frac{\cos 3z}{3!}+\dots ,S=\frac{\sin z}{1!}+\frac{\sin 2z}{2!}+\frac{\sin 3z}{3!}+\dots .$$

[Here $$\begin{array}{rl}C+iS& =1+{\displaystyle \frac{\exp (iz)}{1!}}+{\displaystyle \frac{\exp (2iz)}{2!}}+\cdots =\exp \{\exp (iz)\}\\ & =\exp (\cos z)\{\cos (\sin z)+i\sin (\sin z)\},\end{array}$$ and similarly $$C–iS=\exp \{\exp (-iz)\}=\exp (\cos z)\{\cos (\sin z)–i\sin (\sin z)\}.$$ Hence $$C=\exp (\cos z)\cos (\sin z),S=\exp (\cos z)\sin (\sin z).]$$

10. Sum $$1+\frac{a\cos z}{1!}+\frac{a^2\cos 2z}{2!}+\dots ,\frac{a\sin z}{1!}+\frac{a^2\sin 2z}{2!}+\dots .$$

11. Sum $$1–\frac{\cos 2z}{2!}+\frac{\cos 4z}{4!}–\dots ,\frac{\cos z}{1!}–\frac{\cos 3z}{3!}+\dots$$ and the corresponding series involving sines.

12. Show that $$1+\frac{\cos 4z}{4!}+\frac{\cos 8z}{8!}+\cdots =\frac{1}{2}\{\cos (\cos z)\cosh (\sin z)+\cos (\sin z)\cosh (\cos z)\}.$$

13. Show that the expansions of $\cos (x+h)$ and $\sin (x+h)$ in powers of $h$ (Ex. LVI. 1) are valid for all values of $x$ and $h$, real or complex.