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4.3 Exercises

From Förberedande kurs i matematik 1

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|width="33%" | <math>\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}</math>
|width="33%" | <math>\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:1|Lösning a |Lösning 4.3:1a|Lösning b |Lösning 4.3:1b|Lösning c |Lösning 4.3:1c}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:1|Solution a |Lösning 4.3:1a|Solution b |Lösning 4.3:1b|Solution c |Lösning 4.3:1c}}
===Exercise 4.3:2===
===Exercise 4.3:2===
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|width="50%" | <math>\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}</math>
|width="50%" | <math>\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:2|Lösning a |Lösning 4.3:2a|Lösning b |Lösning 4.3:2b}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:2|Solution a |Lösning 4.3:2a|Solution b |Lösning 4.3:2b}}
===Exercise 4.3:3===
===Exercise 4.3:3===
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|width="50%" | <math>\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}</math>
|width="50%" | <math>\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:3|Lösning a |Lösning 4.3:3a|Lösning b |Lösning 4.3:3b|Lösning c |Lösning 4.3:3c|Lösning d |Lösning 4.3:3d|Lösning e |Lösning 4.3:3e|Lösning f |Lösning 4.3:3f}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:3|Solution a |Lösning 4.3:3a|Solution b |Lösning 4.3:3b|Solution c |Lösning 4.3:3c|Solution d |Lösning 4.3:3d|Solution e |Lösning 4.3:3e|Solution f |Lösning 4.3:3f}}
===Exercise 4.3:4===
===Exercise 4.3:4===
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|width="50%" | <math>\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}</math>
|width="50%" | <math>\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:4|Lösning a |Lösning 4.3:4a|Lösning b |Lösning 4.3:4b|Lösning c |Lösning 4.3:4c|Lösning d |Lösning 4.3:4d|Lösning e |Lösning 4.3:4e|Lösning f |Lösning 4.3:4f}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:4|Solution a |Lösning 4.3:4a|Solution b |Lösning 4.3:4b|Solution c |Lösning 4.3:4c|Solution d |Lösning 4.3:4d|Solution e |Lösning 4.3:4e|Solution f |Lösning 4.3:4f}}
===Exercise 4.3:5===
===Exercise 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>.
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</div>{{#NAVCONTENT:Svar|Svar 4.3:5|Lösning |Lösning 4.3:5}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:5|Solution |Lösning 4.3:5}}
===Exercise 4.3:6===
===Exercise 4.3:6===
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|width="100%" | Determine <math>\ \sin{v}\ </math> and <math>\ \cos{v}\ </math> if <math>\ \tan{v}=3\ </math> and <math>\ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,</math>.
|width="100%" | Determine <math>\ \sin{v}\ </math> and <math>\ \cos{v}\ </math> if <math>\ \tan{v}=3\ </math> and <math>\ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,</math>.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:6|Lösning a |Lösning 4.3:6a|Lösning b |Lösning 4.3:6b|Lösning c |Lösning 4.3:6c}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:6|Solution a |Lösning 4.3:6a|Solution b |Lösning 4.3:6b|Solution c |Lösning 4.3:6c}}
===Exercise 4.3:7===
===Exercise 4.3:7===
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|width="100%" | <math>\cos{x}=\displaystyle \frac{2}{5}\,</math>, <math>\ \cos{y}=\displaystyle \frac{3}{5}\ </math> and <math>\,x\,</math>, <math>\,y\,</math> are angles in the first quadrant.
|width="100%" | <math>\cos{x}=\displaystyle \frac{2}{5}\,</math>, <math>\ \cos{y}=\displaystyle \frac{3}{5}\ </math> and <math>\,x\,</math>, <math>\,y\,</math> are angles in the first quadrant.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:7|Lösning a |Lösning 4.3:7a|Lösning b |Lösning 4.3:7b}}
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</div>{{#NAVCONTENT:Answer|Svar 4.3:7|Solution a |Lösning 4.3:7a|Solution b |Lösning 4.3:7b}}
===Exercise 4.3:8===
===Exercise 4.3:8===
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|width="100%" | <math>\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v</math>
|width="100%" | <math>\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v</math>
|}
|}
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</div>{{#NAVCONTENT:Lösning a |Lösning 4.3:8a|Lösning b |Lösning 4.3:8b|Lösning c |Lösning 4.3:8c|Lösning d |Lösning 4.3:8d}}
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</div>{{#NAVCONTENT:Solution a |Lösning 4.3:8a|Solution b |Lösning 4.3:8b|Solution c |Lösning 4.3:8c|Solution d |Lösning 4.3:8d}}
===Exercise 4.3:9===
===Exercise 4.3:9===
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|width="100%" |(Hint: use the formula for double angles on <math>\,\sin 160^\circ\,</math>.)
|width="100%" |(Hint: use the formula for double angles on <math>\,\sin 160^\circ\,</math>.)
|}
|}
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</div>{{#NAVCONTENT:Lösning |Lösning 4.3:9}}
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</div>{{#NAVCONTENT:Solution |Lösning 4.3:9}}

Revision as of 14:43, 19 August 2008

       Theory          Exercises      

Exercise 4.3:1

Determine the angles v between 2 and 2 which satisfy

a) cosv=cos5 b) sinv=sin7 c) tanv=tan72

Exercise 4.3:2

Determine the angles v between 0 and which satisfy

a) cosv=cos23 b) cosv=cos57

Exercise 4.3:3

Suppose that 2v2 and that sinv=a. With the help of a express

a) sin(v) b) sin(v)
c) cosv d) sin2v 
e) cos2+v  f) sin3+v 

Exercise 4.3:4

Suppose that 0v and that cosv=b. With the help of b express

a) sin2v b) sinv
c) sin2v d) cos2v
e) sinv+4  f) cosv3 

Exercise 4.3:5

Determine cosv and tanv, where v is an acute angle in a triangle such that sinv=75.

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
\displaystyle \cos 20^\circ \cdot \cos 40^\circ \cdot \cos 80^\circ = \displaystyle\frac{1}{8}\,\mbox{.}
(Hint: use the formula for double angles on \displaystyle \,\sin 160^\circ\,.)