# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 18

    Warm-up-exercises

### Exercise

Prove the following properties of the hyperbolic sine and the hyperbolic cosine
    $\cosh x+\sinh x=e^{x}\,.$
    $\cosh x-\sinh x=e^{-x}\,.$
    $(\cosh x)^{2}-(\sinh x)^{2}=1\,.$

### Exercise

Show that the hyperbolic sine is strictly increasing on $\mathbb {R}$.

### Exercise

Prove that the hyperbolic tangent satisfies the following estimate
    $-1\leq \tanh x\leq 1{\text{ for all }}x\in \mathbb {R} .$

### Exercise

Prove by elementary geometric considerations the Sine theorem, i.e. the statement that in a triangle the equalities
    ${\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}\,$
hold, where $a,b,c$ are the side lengths of the edges and $\alpha ,\beta ,\gamma$ are respectively the opposite angles.

### Exercise

Compute the determinants of plane and spatial rotations.

### Exercise

Prove the addition theorems for sine and cosine, using the rotation matrices.

### Exercise

We look at a clock with minute and second hands, both moving continuously. Determine a formula which calculates the angular position of the second hand from the angular position of the minute hand (each starting from the 12-clock-position measured in the clockwise direction).

### Exercise

Prove that the series
    $\sum _{n=1}^{\infty }{\frac {\sin n}{n^{2}}}$
converges.

### Exercise

Determine the coefficients up to $z^{6}$ in the series product $\sum _{n=0}^{\infty }c_{n}z^{n}$ of the sine series and the cosine series.

### Exercise

Let
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
be a periodic function and
    $g\colon \mathbb {R} \longrightarrow \mathbb {R}$
any function. a) Prove that the composite function $g\circ f$ is also periodic. b) Prove that the composite function $f\circ g$ does not need to be periodic.

### Exercise

Let
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
be a continuous periodic function. Prove that $f$ is bounded.
    Hand-in-exercises

### Exercise (3 marks)

Prove that in the power series $\sum _{n=0}^{\infty }c_{n}x^{n}$ of the hyperbolic cosine the coefficients $c_{n}$ are $0$ if $n$ is odd.

### Exercise (3 marks)

Prove that the hyperbolic cosine is strictly decreasing on $\mathbb {R} _{\leq 0}$ and strictly increasing on $\mathbb {R} _{\geq 0}$.
===Exercise Let
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$
be the rotation of the $3$-space around the $z$-axis by $45$ degree counter-clockwise. How does the describing matrix look like with respect to the basis
    ${\begin{pmatrix}1\\2\\4\end{pmatrix}},\,{\begin{pmatrix}3\\3\\-1\end{pmatrix}},\,{\begin{pmatrix}5\\0\\7\end{pmatrix}}?$
Let
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$
be the rotation of the $3$-space around the $z$-axis by $45$ degree counter-clockwise. How does the describing matrix look like with respect to the basis
    ${\begin{pmatrix}1\\2\\4\end{pmatrix}},\,{\begin{pmatrix}3\\3\\-1\end{pmatrix}},\,{\begin{pmatrix}5\\0\\7\end{pmatrix}}?$

### Exercise (5 marks)

Prove the addition theorem
    $\sin(x+y)=\sin x\cdot \cos y+\cos x\cdot \sin y\,$
for the sine using the defining power series.

### Exercise (4 marks)

Let
    $f_{1},f_{2}\colon \mathbb {R} \longrightarrow \mathbb {R}$
be periodic functions with periods respectively $L_{1}$ and $L_{2}$. The quotient $L_{1}/L_{2}$ is a rational number. Prove that $f_{1}+f_{2}$ is also a periodic function.

### Exercise (5 marks)

Consider $n$ complex numbers $z_{1},z_{2},\ldots ,z_{n}$ lying in the disc $B$ with center $(0,0)$ and radius $1$, that is in $B={\left\{z\in \mathbb {C} \mid \vert {z}\vert \leq 1\right\}}$. Prove that there exists a point $w\in B$ such that
    $\sum _{i=1}^{n}\vert {z_{i}-w}\vert \geq n\,.$