4.3 Übungen
Aus Online Mathematik Brückenkurs 1
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Version vom 10:34, 10. Sep. 2008
Exercise 4.3:1
Determine the angles \displaystyle \,v\, between \displaystyle \,\displaystyle \frac{\pi}{2}\, and \displaystyle \,2\pi\, which satisfy
| a) | \displaystyle \cos{v}=\cos{\displaystyle \frac{\pi}{5}} | b) | \displaystyle \sin{v}=\sin{\displaystyle \frac{\pi}{7}} | c) | \displaystyle \tan{v}=\tan{\displaystyle \frac{2\pi}{7}} |
Laden...Exercise 4.3:2
Determine the angles \displaystyle \,v\, between 0 and \displaystyle \,\pi\, which satisfy
| a) | \displaystyle \cos{v} = \cos{\displaystyle \frac{3\pi}{2}} | b) | \displaystyle \cos{v} = \cos{ \displaystyle \frac{7\pi}{5}} |
Exercise 4.3:3
Suppose that \displaystyle \,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\, and that \displaystyle \,\sin{v} = a\,. With the help of \displaystyle \,a express
| a) | \displaystyle \sin{(-v)} | b) | \displaystyle \sin{(\pi-v)} |
| c) | \displaystyle \cos{v} | d) | \displaystyle \sin{\left(\displaystyle \frac{\pi}{2}-v\right)} |
| e) | \displaystyle \cos{\left( \displaystyle \frac{\pi}{2} + v\right)} | f) | \displaystyle \sin{\left( \displaystyle \frac{\pi}{3} + v \right)} |
Exercise 4.3:4
Suppose that \displaystyle \,0 \leq v \leq \pi\, and that \displaystyle \,\cos{v}=b\,. With the help of \displaystyle \,b express
| a) | \displaystyle \sin^2{v} | b) | \displaystyle \sin{v} |
| c) | \displaystyle \sin{2v} | d) | \displaystyle \cos{2v} |
| e) | \displaystyle \sin{\left( v+\displaystyle \frac{\pi}{4} \right)} | f) | \displaystyle \cos{\left( v-\displaystyle \frac{\pi}{3} \right)} |
Exercise 4.3:5
Determine \displaystyle \,\cos{v}\, and \displaystyle \,\tan{v}\,, where \displaystyle \,v\, is an acute angle in a triangle such that \displaystyle \,\sin{v}=\displaystyle \frac{5}{7}\,.
Exercise 4.3:6
| a) | Determine \displaystyle \ \sin{v}\ and \displaystyle \ \tan{v}\ if \displaystyle \ \cos{v}=\displaystyle \frac{3}{4}\ and \displaystyle \ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,. |
| b) | Determine \displaystyle \ \cos{v}\ and \displaystyle \ \tan{v}\ if \displaystyle \ \sin{v}=\displaystyle \frac{3}{10}\ and \displaystyle \,v\, lies in the second quadrant. |
| c) | Determine \displaystyle \ \sin{v}\ and \displaystyle \ \cos{v}\ if \displaystyle \ \tan{v}=3\ and \displaystyle \ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,. |
Exercise 4.3:7
Determine \displaystyle \ \sin{(x+y)}\ if
| a) | \displaystyle \sin{x}=\displaystyle \frac{2}{3}\,,\displaystyle \ \sin{y}=\displaystyle \frac{1}{3}\ and \displaystyle \,x\,, \displaystyle \,y\, are angles in the first quadrant. |
| b) | \displaystyle \cos{x}=\displaystyle \frac{2}{5}\,, \displaystyle \ \cos{y}=\displaystyle \frac{3}{5}\ and \displaystyle \,x\,, \displaystyle \,y\, are angles in the first quadrant. |
Exercise 4.3:8
Show the following trigonometric relations
| a) | \displaystyle \tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v} |
| b) | \displaystyle \displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v} |
| c) | \displaystyle \tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u} |
| d) | \displaystyle \displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v |
Exercise 4.3:9
| Show Feynman's equality | |
| (Hint: use the formula for double angles on \displaystyle \,\sin 160^\circ\,.) |
