4.2 Övningar

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{{Mall:Ej vald flik|[[4.2 Funktioner|Teori]]}}
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{{Ej vald flik|[[4.2 Trigonometriska funktioner|Theory]]}}
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{{Mall:Vald flik|[[4.2 Övningar|Övningar]]}}
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{{Vald flik|[[4.2 Övningar|Exercises]]}}
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===Övning 4.2:1===
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===Exercise 4.2:1===
<div class="ovning">
<div class="ovning">
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Bestäm längden av sidan som är markerad med <math>\,x\,</math> uttryckt med hjälp av de trigonometriska funktionerna.
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Using the trigonometric functions, determine the length of the side marked<math>\,x\,</math>
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{| width="100%" cellspacing="10px"
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 50° och sidor x och 19}}
{{:4.2 - Figur - Rätvinklig triangel med vinkeln 50° och sidor x och 19}}
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|}
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</div>{{#NAVCONTENT:Svar|Svar 4.2:1|Lösning a |Lösning 4.2:1a|Lösning b |Lösning 4.2:1b|Lösning c |Lösning 4.2:1c|Lösning d |Lösning 4.2:1d|Lösning e |Lösning 4.2:1e|Lösning f |Lösning 4.2:1f}}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:1|Solution a |Lösning 4.2:1a|Solution b |Lösning 4.2:1b|Solution c |Lösning 4.2:1c|Solution d |Lösning 4.2:1d|Solution e |Lösning 4.2:1e|Solution f |Lösning 4.2:1f}}
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===Övning 4.2:2===
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===Exercise 4.2:2===
<div class="ovning">
<div class="ovning">
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Bestäm längden av sidan som är markerad med <math>\,x\,</math> uttryckt med hjälp av de trigonometriska funktionerna.
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Determine a trigonometric equation that is satisfied by <math>\,v\,</math>.
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
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|width="50%" | {{:4.2 - Figur - Likbent triangel med toppvinkeln v och sidor 2, 3 och 3}}
|width="50%" | {{:4.2 - Figur - Likbent triangel med toppvinkeln v och sidor 2, 3 och 3}}
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|}
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</div>{{#NAVCONTENT:Svar|Svar 4.2:2|Lösning a |Lösning 4.2:2a|Lösning b |Lösning 4.2:2b|Lösning c |Lösning 4.2:2c|Lösning d |Lösning 4.2:2d|Lösning e |Lösning 4.2:2e|Lösning f |Lösning 4.2:2f}}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:2|Solution a |Lösning 4.2:2a|Solution b |Lösning 4.2:2b|Solution c |Lösning 4.2:2c|Solution d |Lösning 4.2:2d|Solution e |Lösning 4.2:2e|Solution f |Lösning 4.2:2f}}
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===Exercise 4.2:3===
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<div class="ovning">
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Determine
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{| width="100%" cellspacing="10px"
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|a)
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|width="33%" | <math>\sin{\left(-\displaystyle \frac{\pi}{2}\right)}</math>
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|b)
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|width="33%" | <math>\cos{2\pi}</math>
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|c)
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|width="33%" | <math>\sin{9\pi}</math>
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|-
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|d)
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|width="33%" | <math>\cos{\displaystyle \frac{7\pi}{2}}</math>
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|e)
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|width="33%" | <math>\sin{\displaystyle \frac{3\pi}{4}}</math>
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|f)
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|width="33%" | <math>\cos{\left(-\displaystyle \frac{\pi}{6}\right)}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:3|Solution a |Lösning 4.2:3a|Solution b |Lösning 4.2:3b|Solution c |Lösning 4.2:3c|Solution d |Lösning 4.2:3d|Solution e |Lösning 4.2:3e|Solution f |Lösning 4.2:3f}}
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===Exercise 4.2:4===
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<div class="ovning">
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Determine
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{| width="100%" cellspacing="10px"
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|a)
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|width="33%" | <math>\cos{\displaystyle \frac{11\pi}{6}}</math>
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|b)
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|width="33%" | <math>\cos{\displaystyle \frac{11\pi}{3}}</math>
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|c)
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|width="33%" | <math>\tan{\displaystyle \frac{3\pi}{4}}</math>
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|-
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|d)
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|width="33%" | <math>\tan{\pi}</math>
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|e)
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|width="33%" | <math>\tan{\displaystyle \frac{7\pi}{6}}</math>
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|f)
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|width="33%" | <math>\tan{\left(-\displaystyle \frac{5\pi}{3}\right)}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:4|Solution a |Lösning 4.2:4a|Solution b |Lösning 4.2:4b|Solution c |Lösning 4.2:4c|Solution d |Lösning 4.2:4d|Solution e |Lösning 4.2:4e|Solution f |Lösning 4.2:4f}}
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===Exercise 4.2:5===
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<div class="ovning">
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Determine
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{| width="100%" cellspacing="10px"
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|a)
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|width="25%" | <math>\cos{135^\circ}</math>
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|b)
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|width="25%" | <math>\tan{225^\circ}</math>
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|c)
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|width="25%" | <math>\cos{330^\circ}</math>
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|d)
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|width="25%" | <math>\tan{495^\circ}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:5|Solution a |Lösning 4.2:5a|Solution b |Lösning 4.2:5b|Solution c |Lösning 4.2:5c|Solution d |Lösning 4.2:5d}}
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===Exercise 4.2:6===
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<div class="ovning">
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Determine the length of the side marked <math>\,x\,</math>.
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{| width="100%" cellspacing="10px"
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|
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|width="100%" | <center> {{:4.2 - Figur - Två trianglar med vinklar 45° resp. 60° och höjdskillnad x}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:6|Solution |Lösning 4.2:6}}
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===Exercise 4.2:7===
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<div class="ovning">
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In order to determine the width of a river, we measure from two points, A and B on one side of the straight bank to a tree, C, on the opposite side. How wide is the river if the measurements in the figure are correct?
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{| width="100%" cellspacing="10px"
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|
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|width="100%" | <center> {{:4.2 - Figur - Älv}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:7|Solution |Lösning 4.2:7}}
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===Exercise 4.2:8===
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<div class="ovning">
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A rod of length <math>\,\ell\,</math> hangs from two ropes of length <math>\,a\,</math> and <math>\,b\,</math> as shown in the figure. The ropes make angles <math>\,\alpha\,</math> and <math>\,\beta\,</math> with the vertical. Determine a trigonometric equation
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for the angle <math>\,\gamma\,</math> which the rod makes with the vertical.
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{| width="100%" cellspacing="10px"
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|
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|width="100%" | <center> {{:4.2 - Figur - Hängande stång}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:8|Solution |Lösning 4.2:8}}
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===Exercise 4.2:9===
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<div class="ovning">
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The road from ''A'' to ''B'' consists of three straight parts ''AP'', ''PQ'' and ''QB'', which are 4.0 km, 12.0 km and 5.0 km respectively. The angles marked at ''P'' and ''Q'' in the figure are 30° and 90° respectively. Calculate the distance as the crow flies from ''A'' to ''B''. (The exercise is taken from the Swedish National Exam in Mathematics, November 1976, although slightly modified.)
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{| width="100%" cellspacing="10px"
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|
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|width="100%" | <center> {{:4.2 - Figur - Bilväg från A till B via P och Q}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Svar 4.2:9|Solution |Lösning 4.2:9}}
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{{:4.2 - Figur - Två trianglar med vinklar 45° resp. 60° och höjdskillnad x}}
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{{:4.2 - Figur - Älv}}
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{{:4.2 - Figur - Hängande stång}}
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{{:4.2 - Figur - Bilväg från A till B via P och Q}}
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Aktuelle Version

       Theory          Exercises      

Exercise 4.2:1

Using the trigonometric functions, determine the length of the side marked\displaystyle \,x\,

a)

[Image]

b)

[Image]

c)

[Image]

d)

[Image]

e)

[Image]

f)

[Image]

Answer
Svar 4.2:1
Solution a
Lösning 4.2:1a
Solution b
Lösning 4.2:1b
Solution c
Lösning 4.2:1c
Solution d
Lösning 4.2:1d
Solution e
Lösning 4.2:1e
Solution f
Lösning 4.2:1f

Exercise 4.2:2

Determine a trigonometric equation that is satisfied by \displaystyle \,v\,.

a)

[Image]

b)

[Image]

c)

[Image]

d)

[Image]

e)

[Image]

f)

[Image]

Answer
Svar 4.2:2
Solution a
Lösning 4.2:2a
Solution b
Lösning 4.2:2b
Solution c
Lösning 4.2:2c
Solution d
Lösning 4.2:2d
Solution e
Lösning 4.2:2e
Solution f
Lösning 4.2:2f

Exercise 4.2:3

Determine

a) \displaystyle \sin{\left(-\displaystyle \frac{\pi}{2}\right)} b) \displaystyle \cos{2\pi} c) \displaystyle \sin{9\pi}
d) \displaystyle \cos{\displaystyle \frac{7\pi}{2}} e) \displaystyle \sin{\displaystyle \frac{3\pi}{4}} f) \displaystyle \cos{\left(-\displaystyle \frac{\pi}{6}\right)}
Answer
Svar 4.2:3
Solution a
Lösning 4.2:3a
Solution b
Lösning 4.2:3b
Solution c
Lösning 4.2:3c
Solution d
Lösning 4.2:3d
Solution e
Lösning 4.2:3e
Solution f
Lösning 4.2:3f

Exercise 4.2:4

Determine

a) \displaystyle \cos{\displaystyle \frac{11\pi}{6}} b) \displaystyle \cos{\displaystyle \frac{11\pi}{3}} c) \displaystyle \tan{\displaystyle \frac{3\pi}{4}}
d) \displaystyle \tan{\pi} e) \displaystyle \tan{\displaystyle \frac{7\pi}{6}} f) \displaystyle \tan{\left(-\displaystyle \frac{5\pi}{3}\right)}
Answer
Svar 4.2:4
Solution a
Lösning 4.2:4a
Solution b
Lösning 4.2:4b
Solution c
Lösning 4.2:4c
Solution d
Lösning 4.2:4d
Solution e
Lösning 4.2:4e
Solution f
Lösning 4.2:4f

Exercise 4.2:5

Determine

a) \displaystyle \cos{135^\circ} b) \displaystyle \tan{225^\circ} c) \displaystyle \cos{330^\circ} d) \displaystyle \tan{495^\circ}
Answer
Svar 4.2:5
Solution a
Lösning 4.2:5a
Solution b
Lösning 4.2:5b
Solution c
Lösning 4.2:5c
Solution d
Lösning 4.2:5d

Exercise 4.2:6

Determine the length of the side marked \displaystyle \,x\,.

[Image]

Answer
Svar 4.2:6
Solution
Lösning 4.2:6

Exercise 4.2:7

In order to determine the width of a river, we measure from two points, A and B on one side of the straight bank to a tree, C, on the opposite side. How wide is the river if the measurements in the figure are correct?

[Image]

Answer
Svar 4.2:7
Solution
Lösning 4.2:7

Exercise 4.2:8

A rod of length \displaystyle \,\ell\, hangs from two ropes of length \displaystyle \,a\, and \displaystyle \,b\, as shown in the figure. The ropes make angles \displaystyle \,\alpha\, and \displaystyle \,\beta\, with the vertical. Determine a trigonometric equation for the angle \displaystyle \,\gamma\, which the rod makes with the vertical.

[Image]

Answer
Svar 4.2:8
Solution
Lösning 4.2:8

Exercise 4.2:9

The road from A to B consists of three straight parts AP, PQ and QB, which are 4.0 km, 12.0 km and 5.0 km respectively. The angles marked at P and Q in the figure are 30° and 90° respectively. Calculate the distance as the crow flies from A to B. (The exercise is taken from the Swedish National Exam in Mathematics, November 1976, although slightly modified.)

[Image]

Answer
Svar 4.2:9
Solution
Lösning 4.2:9
Von „http://wiki.sommarmatte.se/wikis/2008/bridgecourse1-TU-Berlin/index.php/4.2_%C3%96vningar“