4.4 Exercises

From Förberedande kurs i matematik 1

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{{Ej vald flik|[[4.4 Trigonometriska ekvationer|Teori]]}}
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{{Not selected tab|[[4.4 Trigonometric equations|Theory]]}}
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{{Vald flik|[[4.4 Övningar|Övningar]]}}
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{{Selected tab|[[4.4 Exercises|Exercises]]}}
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===Övning 4.4:1===
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===Exercise 4.4:1===
<div class="ovning">
<div class="ovning">
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För vilka vinklar <math>\,v\,</math>, där <math>\,0 \leq v\leq 2\pi\,</math>, gäller att
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For which angles <math>\,v\,</math>, where <math>\,0 \leq v\leq 2\pi\,</math>, does
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|width="50%" | <math>\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}</math>
|width="50%" | <math>\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:1|Lösning a |Lösning 4.4:1a|Lösning b |Lösning 4.4:1b|Lösning c |Lösning 4.4:1c|Lösning d |Lösning 4.4:1d|Lösning e |Lösning 4.4:1e|Lösning f |Lösning 4.4:1f|Lösning g |Lösning 4.4:1g}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:1|Solution a |Solution 4.4:1a|Solution b |Solution 4.4:1b|Solution c |Solution 4.4:1c|Solution d |Solution 4.4:1d|Solution e |Solution 4.4:1e|Solution f |Solution 4.4:1f|Solution g |Solution 4.4:1g}}
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===Övning 4.4:2===
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===Exercise 4.4:2===
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<div class="ovning">
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L&ouml;s ekvationen
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Solve the equation
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|width="33%" | <math>\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}</math>
|width="33%" | <math>\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:2|Lösning a |Lösning 4.4:2a|Lösning b |Lösning 4.4:2b|Lösning c |Lösning 4.4:2c|Lösning d |Lösning 4.4:2d|Lösning e |Lösning 4.4:2e|Lösning f |Lösning 4.4:2f}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:2|Solution a |Solution 4.4:2a|Solution b |Solution 4.4:2b|Solution c |Solution 4.4:2c|Solution d |Solution 4.4:2d|Solution e |Solution 4.4:2e|Solution f |Solution 4.4:2f}}
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===Övning 4.4:3===
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===Exercise 4.4:3===
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Lös ekvationen
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Solve the equation
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|width="50%" | <math>\sin{3x}=\sin{15^\circ}</math>
|width="50%" | <math>\sin{3x}=\sin{15^\circ}</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:3|Lösning a |Lösning 4.4:3a|Lösning b |Lösning 4.4:3b|Lösning c |Lösning 4.4:3c|Lösning d |Lösning 4.4:3d}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:3|Solution a |Solution 4.4:3a|Solution b |Solution 4.4:3b|Solution c |Solution 4.4:3c|Solution d |Solution 4.4:3d}}
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===Övning 4.4:4===
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===Exercise 4.4:4===
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<div class="ovning">
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Bestäm de vinklar <math>\,v\,</math> i intervallet <math>\,0^\circ \leq v \leq 360^\circ\,</math> som uppfyller <math>\ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,</math>.
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Determine the angles <math>\,v\,</math> in the interval <math>\,0^\circ \leq v \leq 360^\circ\,</math> which satisfy <math>\ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,</math>.
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</div>{{#NAVCONTENT:Svar|Svar 4.4:4|Lösning |Lösning 4.4:4}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:4|Solution |Solution 4.4:4}}
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===Övning 4.4:5===
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===Exercise 4.4:5===
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Lös ekvationen
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Solve the equation
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|width="50%" | <math>\cos{5x}=\cos(x+\pi/5)</math>
|width="50%" | <math>\cos{5x}=\cos(x+\pi/5)</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:5|Lösning a |Lösning 4.4:5a|Lösning b |Lösning 4.4:5b|Lösning c |Lösning 4.4:5c}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:5|Solution a |Solution 4.4:5a|Solution b |Solution 4.4:5b|Solution c |Solution 4.4:5c}}
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===Övning 4.4:6===
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===Exercise 4.4:6===
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Lös ekvationen
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Solve the equation
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|width="50%" | <math>\sin 2x = -\sin x</math>
|width="50%" | <math>\sin 2x = -\sin x</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:6|Lösning a |Lösning 4.4:6a|Lösning b |Lösning 4.4:6b|Lösning c |Lösning 4.4:6c}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:6|Solution a |Solution 4.4:6a|Solution b |Solution 4.4:6b|Solution c |Solution 4.4:6c}}
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===Övning 4.4:7===
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===Exercise 4.4:7===
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Lös ekvationen
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Solve the equation
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|width="50%" | <math>\cos{3x}=\sin{4x}</math>
|width="50%" | <math>\cos{3x}=\sin{4x}</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:7|Lösning a |Lösning 4.4:7a|Lösning b |Lösning 4.4:7b|Lösning c |Lösning 4.4:7c}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:7|Solution a |Solution 4.4:7a|Solution b |Solution 4.4:7b|Solution c |Solution 4.4:7c}}
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===Övning 4.4:8===
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===Exercise 4.4:8===
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Lös ekvationen
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Solve the equation
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|width="50%" | <math>\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}</math>
|width="50%" | <math>\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}</math>
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</div>{{#NAVCONTENT:Svar|Svar 4.4:8|Lösning a |Lösning 4.4:8a|Lösning b |Lösning 4.4:8b|Lösning c |Lösning 4.4:8c}}
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</div>{{#NAVCONTENT:Answer|Answer 4.4:8|Solution a |Solution 4.4:8a|Solution b |Solution 4.4:8b|Solution c |Solution 4.4:8c}}

Current revision

       Theory          Exercises      

Exercise 4.4:1

For which angles \displaystyle \,v\,, where \displaystyle \,0 \leq v\leq 2\pi\,, does

a) \displaystyle \sin{v}=\displaystyle \frac{1}{2} b) \displaystyle \cos{v}=\displaystyle \frac{1}{2}
c) \displaystyle \sin{v}=1 d) \displaystyle \tan{v}=1
e) \displaystyle \cos{v}=2 f) \displaystyle \sin{v}=-\displaystyle \frac{1}{2}
g) \displaystyle \tan{v}=-\displaystyle \frac{1}{\sqrt{3}}
Answer
Answer 4.4:1
Solution a
Solution 4.4:1a
Solution b
Solution 4.4:1b
Solution c
Solution 4.4:1c
Solution d
Solution 4.4:1d
Solution e
Solution 4.4:1e
Solution f
Solution 4.4:1f
Solution g
Solution 4.4:1g

Exercise 4.4:2

Solve the equation

a) \displaystyle \sin{x}=\displaystyle \frac{\sqrt{3}}{2} b) \displaystyle \cos{x}=\displaystyle \frac{1}{2} c) \displaystyle \sin{x}=0
d) \displaystyle \sin{5x}=\displaystyle \frac{1}{\sqrt{2}} e) \displaystyle \sin{5x}=\displaystyle \frac{1}{2} f) \displaystyle \cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}
Answer
Answer 4.4:2
Solution a
Solution 4.4:2a
Solution b
Solution 4.4:2b
Solution c
Solution 4.4:2c
Solution d
Solution 4.4:2d
Solution e
Solution 4.4:2e
Solution f
Solution 4.4:2f

Exercise 4.4:3

Solve the equation

a) \displaystyle \cos{x}=\cos{\displaystyle \frac{\pi}{6}} b) \displaystyle \sin{x}=\sin{\displaystyle \frac{\pi}{5}}
c) \displaystyle \sin{(x+40^\circ)}=\sin{65^\circ} d) \displaystyle \sin{3x}=\sin{15^\circ}
Answer
Answer 4.4:3
Solution a
Solution 4.4:3a
Solution b
Solution 4.4:3b
Solution c
Solution 4.4:3c
Solution d
Solution 4.4:3d

Exercise 4.4:4

Determine the angles \displaystyle \,v\, in the interval \displaystyle \,0^\circ \leq v \leq 360^\circ\, which satisfy \displaystyle \ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,.

Answer
Answer 4.4:4
Solution
Solution 4.4:4


Exercise 4.4:5

Solve the equation

a) \displaystyle \sin{3x}=\sin{x} b) \displaystyle \tan{x}=\tan{4x}
c) \displaystyle \cos{5x}=\cos(x+\pi/5)
Answer
Answer 4.4:5
Solution a
Solution 4.4:5a
Solution b
Solution 4.4:5b
Solution c
Solution 4.4:5c

Exercise 4.4:6

Solve the equation

a) \displaystyle \sin x\cdot \cos 3x = 2\sin x b) \displaystyle \sqrt{2}\sin{x}\cos{x}=\cos{x}
c) \displaystyle \sin 2x = -\sin x
Answer
Answer 4.4:6
Solution a
Solution 4.4:6a
Solution b
Solution 4.4:6b
Solution c
Solution 4.4:6c

Exercise 4.4:7

Solve the equation

a) \displaystyle 2\sin^2{x}+\sin{x}=1 b) \displaystyle 2\sin^2{x}-3\cos{x}=0
c) \displaystyle \cos{3x}=\sin{4x}
Answer
Answer 4.4:7
Solution a
Solution 4.4:7a
Solution b
Solution 4.4:7b
Solution c
Solution 4.4:7c

Exercise 4.4:8

Solve the equation

a) \displaystyle \sin{2x}=\sqrt{2}\cos{x} b) \displaystyle \sin{x}=\sqrt{3}\cos{x}
c) \displaystyle \displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}
Answer
Answer 4.4:8
Solution a
Solution 4.4:8a
Solution b
Solution 4.4:8b
Solution c
Solution 4.4:8c
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