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4.4 Exercises

From Förberedande kurs i matematik 1

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(Ny sida: __NOTOC__ {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | style="border-bottom:1px solid #000" width="5px" |   {{Mall:Ej vald flik|[[4.4 Trigonometriska ekvati...)
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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
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{{Mall:Ej vald flik|[[4.4 Trigonometriska ekvationer|Teori]]}}
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{{Not selected tab|[[4.4 Trigonometric equations|Theory]]}}
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{{Mall:Vald flik|[[4.4 Övningar|Övningar]]}}
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{{Selected tab|[[4.4 Exercises|Exercises]]}}
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| style="border-bottom:1px solid #000" width="100%"|  
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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
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%" | <math>\cos{v}=2</math>
|width="50%" | <math>\cos{v}=2</math>
|f)
|f)
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|width="50%" | <math>\sin{v}=-\displaystyle \frac{1}{2}$</math>
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|width="50%" | <math>\sin{v}=-\displaystyle \frac{1}{2}</math>
|-
|-
|g)
|g)
|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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===Exercise 4.4:2===
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<div class="ovning">
 +
Solve the equation
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{| width="100%" cellspacing="10px"
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|a)
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|width="33%" | <math>\sin{x}=\displaystyle \frac{\sqrt{3}}{2}</math>
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|b)
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|width="33%" | <math>\cos{x}=\displaystyle \frac{1}{2} </math>
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|c)
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|width="33%" | <math>\sin{x}=0</math>
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|-
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|d)
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|width="33%" | <math>\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} </math>
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|e)
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|width="33%" | <math>\sin{5x}=\displaystyle \frac{1}{2}</math>
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|f)
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|width="33%" | <math>\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}</math>
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|}
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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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===Exercise 4.4:3===
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<div class="ovning">
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Solve the equation
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%" | <math>\cos{x}=\cos{\displaystyle \frac{\pi}{6}}</math>
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|b)
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|width="50%" | <math>\sin{x}=\sin{\displaystyle \frac{\pi}{5}}</math>
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|-
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|c)
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|width="50%" | <math>\sin{(x+40^\circ)}=\sin{65^\circ}</math>
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|d)
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|width="50%" | <math>\sin{3x}=\sin{15^\circ}</math>
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|}
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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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 +
===Exercise 4.4:4===
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<div class="ovning">
 +
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:Answer|Answer 4.4:4|Solution |Solution 4.4:4}}
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===Exercise 4.4:5===
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<div class="ovning">
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Solve the equation
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="50%" | <math>\sin{3x}=\sin{x}</math>
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|b)
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|width="50%" | <math>\tan{x}=\tan{4x}</math>
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|-
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|c)
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|width="50%" | <math>\cos{5x}=\cos(x+\pi/5)</math>
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|}
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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}}
 +
 
 +
===Exercise 4.4:6===
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<div class="ovning">
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Solve the equation
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="50%" | <math>\sin x\cdot \cos 3x = 2\sin x</math>
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|b)
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|width="50%" | <math>\sqrt{2}\sin{x}\cos{x}=\cos{x}</math>
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|-
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|c)
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|width="50%" | <math>\sin 2x = -\sin x</math>
 +
|}
 +
</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}}
 +
 
 +
===Exercise 4.4:7===
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<div class="ovning">
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Solve the equation
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{| width="100%" cellspacing="10px"
 +
|a)
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|width="50%" | <math>2\sin^2{x}+\sin{x}=1</math>
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|b)
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|width="50%" | <math>2\sin^2{x}-3\cos{x}=0</math>
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|-
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|c)
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|width="50%" | <math>\cos{3x}=\sin{4x}</math>
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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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===Exercise 4.4:8===
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<div class="ovning">
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Solve the equation
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%" | <math>\sin{2x}=\sqrt{2}\cos{x}</math>
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|b)
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|width="50%" | <math>\sin{x}=\sqrt{3}\cos{x}</math>
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|-
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|c)
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|width="50%" | <math>\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}</math>
 +
|}
 +
</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 v, where 0v2, does

a) sinv=21 b) cosv=21
c) sinv=1 d) tanv=1
e) cosv=2 f) sinv=−21
g) tanv=−13

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f | Solution g

Exercise 4.4:2

Solve the equation

a) sinx=23  b) cosx=21 c) sinx=0
d) sin5x=12 e) sin5x=21 f) cos3x=−12

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f

Exercise 4.4:3

Solve the equation

a) cosx=cos6 b) sinx=sin5
c) sin(x+40)=sin65 d) sin3x=sin15

Answer | Solution a | Solution b | Solution c | Solution d

Exercise 4.4:4

Determine the angles v in the interval 0v360 which satisfy  cos2v+10=cos110 .

Answer | Solution


Exercise 4.4:5

Solve the equation

a) sin3x=sinx b) tanx=tan4x
c) cos5x=cos(x+5)

Answer | Solution a | Solution b | Solution c

Exercise 4.4:6

Solve the equation

a) sinxcos3x=2sinx b) 2sinxcosx=cosx 
c) sin2x=−sinx

Answer | Solution a | Solution b | Solution c

Exercise 4.4:7

Solve the equation

a) 2sin2x+sinx=1 b) 2sin2x−3cosx=0
c) cos3x=sin4x

Answer | Solution a | Solution b | Solution c

Exercise 4.4:8

Solve the equation

a) sin2x=2cosx  b) sinx=3cosx 
c) 1cos2x=1−tanx

Answer | Solution a | Solution b | Solution c

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