# 4.5 The trigonometric and hyperbolic functions

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Two theorems covering differentiation of trigonometric and hyperbolic functions, including practice exercises corresponding to the theorems.

The laws of exponents and the algebraic connections between the exponential function and the trigonometric andhyperbolic functions, give the following “addition formulas:”

The following identities hold for all complex numbers $z$ and $w.$

$sin\left(z+w\right)=sin\left(z\right)cos\left(w\right)+cos\left(z\right)sin\left(w\right).$
$cos\left(z+w\right)=cos\left(z\right)cos\left(w\right)-sin\left(z\right)sin\left(w\right).$
$sinh\left(z+w\right)=sinh\left(z\right)cosh\left(w\right)+cosh\left(z\right)sinh\left(w\right).$
$cosh\left(z+w\right)=cosh\left(z\right)cosh\left(w\right)+sinh\left(z\right)sinh\left(w\right).$

We derive the first formula and leave the others to an exercise.

First, for any two real numbers $x$ and $y,$ we have

$\begin{array}{ccc}\hfill cos\left(x+y\right)+isin\left(x+y\right)& =& {e}^{i\left(x+y\right)}\hfill \\ & =& {e}^{ix}{e}^{iy}\hfill \\ & =& \left(cosx+isinx\right)×\left(cosy+isiny\right)\hfill \\ & =& cosxcosy-sinxsiny+i\left(cosxsiny+sinxcosy\right),\hfill \end{array}$

which, equating real and imaginary parts, gives that

$cos\left(x+y\right)=cosxcosy-sinxsiny$

and

$sin\left(x+y\right)=sinxcosy+cosxsiny.$

The second of these equations is exactly what we want, but this calculation only shows that it holds for real numbers $x$ and $y.$ We can use the Identity Theorem to show that in fact this formula holds for all complex numbers $z$ and $w.$ Thus, fix a real number $y.$ Let $f\left(z\right)=sinzcosy+coszsiny,$ and let

$g\left(z\right)=sin\left(z+y\right)=\frac{1}{2i}\left({e}^{i\left(z+y\right)}-{e}^{-i\left(z+y\right)}=\frac{1}{2i}\left({e}^{iz}{e}^{iy}-{e}^{-iz}{e}^{-iy}\right).$

Then both $f$ and $g$ are power series functions of the variable $z.$ Furthermore, by the previous calculation, $f\left(1/k\right)=g\left(1/k\right)$ for all positive integers $k.$ Hence, by the Identity Theorem, $f\left(z\right)=g\left(z\right)$ for all complex $z.$ Hence we have the formula we want for all complex numbers $z$ and all real numbers $y.$

To finish the proof, we do the same trick one more time. Fix a complex number $z.$ Let $f\left(w\right)=sinzcosw+coszsinw,$ and let

$g\left(w\right)=sin\left(z+w\right)=\frac{1}{2i}\left({e}^{i\left(z+w\right)}-{e}^{-i\left(z+w\right)}=\frac{1}{2i}\left({e}^{iz}{e}^{iw}-{e}^{-iz}{e}^{-iw}\right).$

Again, both $f$ and $g$ are power series functions of the variable $w,$ and they agree on the sequence $\left\{1/k\right\}.$ Hence they agree everywhere, and this completes the proof of the first addition formula.

1. Derive the remaining three addition formulas of the preceding theorem.
2. From the addition formulas, derive the two “half angle” formulas for the trigonometric functions:
${sin}^{2}\left(z\right)=\frac{1-cos\left(2z\right)}{2},$
and
${cos}^{2}\left(z\right)=\frac{1+cos\left(2z\right)}{2}.$

The trigonometric functions $sin$ and $cos$ are periodic with period $2\pi ;$ i.e., $sin\left(z+2\pi \right)=sin\left(z\right)$ and $cos\left(z+2\pi \right)=cos\left(z\right)$ for all complex numbers $z.$

We have from the preceding exercise that $sin\left(z+2\pi \right)=sin\left(z\right)cos\left(2\pi \right)+cos\left(z\right)sin\left(2\pi \right),$ so that the periodicity assertion for the sine function will follow if we show that $cos\left(2\pi \right)=1$ and $sin\left(2\pi \right)=0.$ From part (b) of the preceding exercise, we have that

$0={sin}^{2}\left(\pi \right)=\frac{1-cos\left(2\pi \right)}{2}$

which shows that $cos\left(2\pi \right)=1.$ Since ${cos}^{2}+{sin}^{2}=1,$ it then follows that $sin\left(2\pi \right)=0.$

The periodicity of the cosine function is proved similarly.

1. Prove that the hyperbolic functions $sinh$ and $cosh$ are periodic. What is the period?
2. Prove that the hyperbolic cosine $cosh\left(x\right)$ is never 0 for $x$ a real number, that the hyperbolic tangent $tanh\left(x\right)=sinh\left(x\right)/cosh\left(x\right)$ is bounded and increasing from $R$ onto $\left(-1,1\right),$ and that the inverse hyperbolic tangent has derivative given by ${{tanh}^{-1}}^{\text{'}}\left(y\right)=1/\left(1-{y}^{2}\right).$
3. Verify that for all $y\in \left(-1,1\right)$
${tanh}^{-1}\left(y\right)=ln\left(\sqrt{\frac{1+y}{1-y}}\right).$

Let $z$ be a nonzero complex number. Prove that there exists a unique real number $0\le \theta <2\pi$ such that $z=r{e}^{i\theta },$ where $r=|z|.$

HINT: If $z=a+bi,$ then $z=r\left(\frac{a}{r}+\frac{b}{r}i.$ Observe that $-1\le \frac{a}{r}\le 1,$ $-1\le \frac{b}{r}\le 1,$ and ${\left(\frac{a}{r}\right)}^{2}+{\left(\frac{b}{r}\right)}^{2}=1.$ Show that there exists a unique $0\le \theta <2\pi$ such that $\frac{a}{r}=cos\theta$ and $\frac{b}{r}=sin\theta .$