4.3 Übungen
Aus Online Mathematik Brückenkurs 1
(Unterschied zwischen Versionen)
K (Robot: Automated text replacement (-Exercises +Übungen)) |
K (Robot: Automated text replacement (-Exercise +Übung)) |
||
| Zeile 7: | Zeile 7: | ||
|} | |} | ||
| - | === | + | ===Übung 4.3:1=== |
<div class="ovning"> | <div class="ovning"> | ||
Determine the angles <math>\,v\,</math> between <math>\,\displaystyle \frac{\pi}{2}\,</math> and <math>\,2\pi\,</math> which satisfy | Determine the angles <math>\,v\,</math> between <math>\,\displaystyle \frac{\pi}{2}\,</math> and <math>\,2\pi\,</math> which satisfy | ||
| Zeile 20: | Zeile 20: | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:1|Solution a |Solution 4.3:1a|Solution b |Solution 4.3:1b|Solution c |Solution 4.3:1c}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:1|Solution a |Solution 4.3:1a|Solution b |Solution 4.3:1b|Solution c |Solution 4.3:1c}} | ||
| - | === | + | ===Übung 4.3:2=== |
<div class="ovning"> | <div class="ovning"> | ||
Determine the angles <math>\,v\,</math> between 0 and <math>\,\pi\,</math> which satisfy | Determine the angles <math>\,v\,</math> between 0 and <math>\,\pi\,</math> which satisfy | ||
| Zeile 31: | Zeile 31: | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:2|Solution a |Solution 4.3:2a|Solution b |Solution 4.3:2b}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:2|Solution a |Solution 4.3:2a|Solution b |Solution 4.3:2b}} | ||
| - | === | + | ===Übung 4.3:3=== |
<div class="ovning"> | <div class="ovning"> | ||
Suppose that <math>\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\,</math> and that <math>\,\sin{v} = a\,</math>. With the help of <math>\,a</math> express | Suppose that <math>\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\,</math> and that <math>\,\sin{v} = a\,</math>. With the help of <math>\,a</math> express | ||
| Zeile 52: | Zeile 52: | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:3|Solution a |Solution 4.3:3a|Solution b |Solution 4.3:3b|Solution c |Solution 4.3:3c|Solution d |Solution 4.3:3d|Solution e |Solution 4.3:3e|Solution f |Solution 4.3:3f}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:3|Solution a |Solution 4.3:3a|Solution b |Solution 4.3:3b|Solution c |Solution 4.3:3c|Solution d |Solution 4.3:3d|Solution e |Solution 4.3:3e|Solution f |Solution 4.3:3f}} | ||
| - | === | + | ===Übung 4.3:4=== |
<div class="ovning"> | <div class="ovning"> | ||
Suppose that <math>\,0 \leq v \leq \pi\,</math> and that <math>\,\cos{v}=b\,</math>. With the help of <math>\,b</math> express | Suppose that <math>\,0 \leq v \leq \pi\,</math> and that <math>\,\cos{v}=b\,</math>. With the help of <math>\,b</math> express | ||
| Zeile 73: | Zeile 73: | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:4|Solution a |Solution 4.3:4a|Solution b |Solution 4.3:4b|Solution c |Solution 4.3:4c|Solution d |Solution 4.3:4d|Solution e |Solution 4.3:4e|Solution f |Solution 4.3:4f}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:4|Solution a |Solution 4.3:4a|Solution b |Solution 4.3:4b|Solution c |Solution 4.3:4c|Solution d |Solution 4.3:4d|Solution e |Solution 4.3:4e|Solution f |Solution 4.3:4f}} | ||
| - | === | + | ===Übung 4.3:5=== |
<div class="ovning"> | <div class="ovning"> | ||
Determine <math>\,\cos{v}\,</math> and <math>\,\tan{v}\,</math>, where <math>\,v\,</math> is an acute angle in a triangle such that <math>\,\sin{v}=\displaystyle \frac{5}{7}\,</math>. | Determine <math>\,\cos{v}\,</math> and <math>\,\tan{v}\,</math>, where <math>\,v\,</math> is an acute angle in a triangle such that <math>\,\sin{v}=\displaystyle \frac{5}{7}\,</math>. | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:5|Solution |Solution 4.3:5}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:5|Solution |Solution 4.3:5}} | ||
| - | === | + | ===Übung 4.3:6=== |
<div class="ovning"> | <div class="ovning"> | ||
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
| Zeile 92: | Zeile 92: | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:6|Solution a |Solution 4.3:6a|Solution b |Solution 4.3:6b|Solution c |Solution 4.3:6c}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:6|Solution a |Solution 4.3:6a|Solution b |Solution 4.3:6b|Solution c |Solution 4.3:6c}} | ||
| - | === | + | ===Übung 4.3:7=== |
<div class="ovning"> | <div class="ovning"> | ||
Determine <math>\ \sin{(x+y)}\ </math> if | Determine <math>\ \sin{(x+y)}\ </math> if | ||
| Zeile 104: | Zeile 104: | ||
</div>{{#NAVCONTENT:Answer|Answer 4.3:7|Solution a |Solution 4.3:7a|Solution b |Solution 4.3:7b}} | </div>{{#NAVCONTENT:Answer|Answer 4.3:7|Solution a |Solution 4.3:7a|Solution b |Solution 4.3:7b}} | ||
| - | === | + | ===Übung 4.3:8=== |
<div class="ovning"> | <div class="ovning"> | ||
Show the following trigonometric relations | Show the following trigonometric relations | ||
| Zeile 122: | Zeile 122: | ||
</div>{{#NAVCONTENT:Solution a |Solution 4.3:8a|Solution b |Solution 4.3:8b|Solution c |Solution 4.3:8c|Solution d |Solution 4.3:8d}} | </div>{{#NAVCONTENT:Solution a |Solution 4.3:8a|Solution b |Solution 4.3:8b|Solution c |Solution 4.3:8c|Solution d |Solution 4.3:8d}} | ||
| - | === | + | ===Übung 4.3:9=== |
<div class="ovning"> | <div class="ovning"> | ||
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
Version vom 09:15, 22. Okt. 2008
Übung 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}} |
Answer
Laden...
Solution a
Solution b
Solution c
Übung 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}} |
Übung 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)} |
Answer
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
Übung 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)} |
Answer
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
Übung 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}\,.
Answer
Solution
Übung 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}\,. |
Answer
Solution a
Solution b
Solution c
Übung 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. |
Übung 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 |
Solution a
Solution b
Solution c
Solution d
Übung 4.3:9
| Show Feynman's equality | |
| (Hint: use the formula for double angles on \displaystyle \,\sin 160^\circ\,.) |
Solution
