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

1. ${}\cosh x+\sinh x=e^{x}\,.$

2. ${}\cosh x-\sinh x=e^{-x}\,.$

3. ${}(\cosh x)^{2}-(\sinh x)^{2}=1\,.$

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).

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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 ( 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 ( marks)

Let

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

be the space rotation by ${}45$ degree aroand the ${}z$-axis counterclockwise. How does the matrix describing ${}\varphi$ with respect to the basis

${\begin{pmatrix}1\\2\\4\end{pmatrix}},\,{\begin{pmatrix}3\\3\\-1\end{pmatrix}},\,{\begin{pmatrix}5\\0\\7\end{pmatrix}}$

look like?

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

${}\sum _{i=1}^{n}\vert {z_{i}-w}\vert \geq n\,.$