4.3 Exercises

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{{Mall:Ej vald flik|[[4.3 Trigonometriska samband|Teori]]}}
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{{Not selected tab|[[4.3 Trigonometric relations|Theory]]}}
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{{Mall:Vald flik|[[4.3 Övningar|Övningar]]}}
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{{Selected tab|[[4.3 Exercises|Exercises]]}}
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===Övning 4.3:1===
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===Exercise 4.3:1===
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<div class="ovning">
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Bestäm de vinklar <math>\,v\,</math> mellan <math>\,\displaystyle \frac{\pi}{2}\,</math> och <math>\,2\pi\,</math> som uppfyller
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Determine the angles <math>\,v\,</math> between <math>\,\displaystyle \frac{\pi}{2}\,</math> and <math>\,2\pi\,</math> which satisfy
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|a)
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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|Answer 4.3:1|Solution a |Solution 4.3:1a|Solution b |Solution 4.3:1b|Solution c |Solution 4.3:1c}}
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===Övning 4.3:2===
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===Exercise 4.3:2===
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<div class="ovning">
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Bestäm de vinklar <math>\,v\,</math> mellan 0 och <math>\,\pi\,</math> som uppfyller
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Determine the angles <math>\,v\,</math> between 0 and <math>\,\pi\,</math> which satisfy
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|a)
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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|Answer 4.3:2|Solution a |Solution 4.3:2a|Solution b |Solution 4.3:2b}}
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===Övning 4.3:3===
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===Exercise 4.3:3===
<div class="ovning">
<div class="ovning">
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Antag att <math>\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\,</math> och att <math>\,\sin{v} = a\,</math>. Uttryck med hjälp av <math>\,a</math>
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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
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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|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}}
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===Övning 4.3:4===
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===Exercise 4.3:4===
<div class="ovning">
<div class="ovning">
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Antag att <math>\,0 \leq v \leq \pi\,</math> och att <math>\,\cos{v}=b\,</math>. Uttryck med hjälp av <math>\,b</math>
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Suppose that <math>\,0 \leq v \leq \pi\,</math> and that <math>\,\cos{v}=b\,</math>. With the help of <math>\,b</math> express
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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|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}}
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===Övning 4.3:5===
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===Exercise 4.3:5===
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<div class="ovning">
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För en spetsig vinkel <math>\,v\,</math> i en triangel gäller att <math>\,\sin{v}=\displaystyle \frac{5}{7}\,</math>. Bestäm <math>\,\cos{v}\,</math> och <math>\,\tan{v}\,</math>.
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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|Answer 4.3:5|Solution |Solution 4.3:5}}
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===Övning 4.3:6===
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===Exercise 4.3:6===
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|a)
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|width="100%" | Bestäm <math>\ \sin{v}\ </math> och <math>\ \tan{v}\ </math> om <math>\ \cos{v}=\displaystyle \frac{3}{4}\ </math> och <math>\ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,</math>.
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|width="100%" | Determine <math>\ \sin{v}\ </math> and <math>\ \tan{v}\ </math> if <math>\ \cos{v}=\displaystyle \frac{3}{4}\ </math> and <math>\ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,</math>.
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|b)
|b)
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|width="100%" | Bestäm <math>\ \cos{v}\ </math> och <math>\ \tan{v}\ </math> om <math>\ \sin{v}=\displaystyle \frac{3}{10}\ </math> och <math>\,v\,</math> ligger i den andra kvadranten.
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|width="100%" | Determine <math>\ \cos{v}\ </math> and <math>\ \tan{v}\ </math> if <math>\ \sin{v}=\displaystyle \frac{3}{10}\ </math> and <math>\,v\,</math> lies in the second quadrant.
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|c)
|c)
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|width="100%" | Bestäm <math>\ \sin{v}\ </math> och <math>\ \cos{v}\ </math> om <math>\ \tan{v}=3\ </math> och <math>\ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,</math>.
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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>.
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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|Answer 4.3:6|Solution a |Solution 4.3:6a|Solution b |Solution 4.3:6b|Solution c |Solution 4.3:6c}}
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===Övning 4.3:7===
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===Exercise 4.3:7===
<div class="ovning">
<div class="ovning">
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Bestäm <math>\ \sin{(x+y)}\ </math> om
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Determine <math>\ \sin{(x+y)}\ </math> if
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | <math>\sin{x}=\displaystyle \frac{2}{3}\,</math>,<math>\ \sin{y}=\displaystyle \frac{1}{3}\ </math> och <math>\,x\,$, $\,y\,</math> är vinklar i första kvadranten..
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|width="100%" | <math>\sin{x}=\displaystyle \frac{2}{3}\,</math>,<math>\ \sin{y}=\displaystyle \frac{1}{3}\ </math> and <math>\,x\,</math>, <math> \,y\,</math> are angles in the first quadrant.
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|-
|b)
|b)
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|width="100%" | <math>\cos{x}=\displaystyle \frac{2}{5}\,</math>, <math>\ \cos{y}=\displaystyle \frac{3}{5}\ </math> och <math>\,x\,</math>, <math>\,y\,</math> är vinklar i första kvadranten.
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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.
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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|Answer 4.3:7|Solution a |Solution 4.3:7a|Solution b |Solution 4.3:7b}}
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===Övning 4.3:8===
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===Exercise 4.3:8===
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Visa f&ouml;ljande trigonometriska samband
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Show the following trigonometric relations
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|width="100%" | <math>\tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}</math>
|width="100%" | <math>\tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}</math>
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|d)
|d)
|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:Svar|Svar 4.3:8|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 |Solution 4.3:8a|Solution b |Solution 4.3:8b|Solution c |Solution 4.3:8c|Solution d |Solution 4.3:8d}}
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===Exercise 4.3:9===
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|
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|width="100%" | Show Feynman's equality
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|width="100%" |<center> <math>\cos 20^\circ \cdot \cos 40^\circ \cdot \cos 80^\circ = \displaystyle\frac{1}{8}\,\mbox{.}</math> </center>
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|width="100%" |(Hint: use the formula for double angles on <math>\,\sin 160^\circ\,</math>.)
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</div>{{#NAVCONTENT:Solution |Solution 4.3:9}}

Current revision

       Theory          Exercises      

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}}
Answer
Answer 4.3:1
Solution a
Solution 4.3:1a
Solution b
Solution 4.3:1b
Solution c
Solution 4.3:1c

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}}
Answer
Answer 4.3:2
Solution a
Solution 4.3:2a
Solution b
Solution 4.3:2b

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)}
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

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)}
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

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}\,.

Answer
Answer 4.3:5
Solution
Solution 4.3:5

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}\,.
Answer
Answer 4.3:6
Solution a
Solution 4.3:6a
Solution b
Solution 4.3:6b
Solution c
Solution 4.3:6c

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.
Answer
Answer 4.3:7
Solution a
Solution 4.3:7a
Solution b
Solution 4.3:7b

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
Solution a
Solution 4.3:8a
Solution b
Solution 4.3:8b
Solution c
Solution 4.3:8c
Solution d
Solution 4.3:8d

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\,.)
Solution
Solution 4.3:9
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