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



Warm-up-exercises

Prove the following properties of the hyperbolic sine and the hyperbolic cosine

  1.  
  2.  
  3.  



Show that the hyperbolic sine is strictly increasing on  .



Prove that the hyperbolic tangent satisfies the following estimate

 



Prove by elementary geometric considerations the Sine theorem, i.e. the statement that in a triangle the equalities

 

hold, where   are the side lengths of the edges and   are respectively the opposite angles.



Compute the determinants of plane and spatial rotations.



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



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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Prove that the series

 

converges.



Determine the coefficients up to   in the series product   of the sine series and the cosine series.



Let

 

be a periodic function and

 

any function. a) Prove that the composite function   is also periodic. b) Prove that the composite function   does not need to be periodic.



Let

 

be a continuous periodic function. Prove that   is bounded.






Hand-in-exercises

Exercise ( marks)

Prove that in the power series   of the hyperbolic cosine the coefficients   are   if   is odd.



Exercise (3 marks)

Prove that the hyperbolic cosine is strictly decreasing on   and strictly increasing on  .



Exercise ( marks)

Let

 

be the space rotation by   degree aroand the  -axis counterclockwise. How does the matrix describing   with respect to the basis

 

look like?



Exercise (5 marks)

Prove the addition theorem

 

for the sine using the defining power series.



Exercise (4 marks)

Let

 

be periodic functions with periods respectively   and  . The quotient   is a rational number. Prove that   is also a periodic function.



Exercise (5 marks)

Consider   complex numbers   lying in the disc   with center   and radius  , that is in  . Prove that there exists a point   such that