4.2 Exercises

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{{Mall:Ej vald flik|[[4.2 Fuktioner|Teori]]}}
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{{Not selected tab|[[4.2 Trigonometric functions|Theory]]}}
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{{Selected tab|[[4.2 Exercises|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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|a)
|a)
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|width="50%" | {{:4.2 - Figur - Rätvinklig triangel med vinkeln 27° och sidor x och 13}}
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle 27° and sides x and 13}}
|b)
|b)
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|width="50%" |
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 32° och sidor x och 25}}
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{{:4.2 - Figure - A right-angled triangle with angle 32° and sides x and 25}}
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|c)
|c)
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|width="50%" | {{:4.2 - Figur - Rätvinklig triangel med vinkeln 40° och sidor 14 och x}}
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle 40° and sides 14 and x}}
|d)
|d)
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|width="50%" | {{:4.2 - Figur - Rätvinklig triangel med vinkeln 20° och sidor 16 och x}}
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle 20° and sides 16 and x}}
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|e)
|e)
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|width="50%" | {{:4.2 - Figur - Rätvinklig triangel med vinkeln 35° och sidor 11 och x}}
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle 35° and sides 11 and x}}
|f)
|f)
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|width="50%" |
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 50° och sidor x och 19}}
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{{:4.2 - Figure - A right-angled triangle with angle 50° and sides x and 19}}
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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|Answer 4.2:1|Solution a |Solution 4.2:1a|Solution b |Solution 4.2:1b|Solution c |Solution 4.2:1c|Solution d |Solution 4.2:1d|Solution e |Solution 4.2:1e|Solution f |Solution 4.2:1f}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 27° och sidor x och 13}}
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===Exercise 4.2:2===
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<div class="ovning">
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Determine a trigonometric equation that is satisfied by <math>\,v\,</math>.
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|a)
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle v and sides 2 and 5}}
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|b)
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle v and sides 70 and 110}}
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|-
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|c)
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle v and sides 5 and 7}}
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|d)
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angle v and sides 3 and 5}}
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|-
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|e)
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|width="50%" | {{:4.2 - Figure - A right-angled triangle with angles v and 60° and side 5}}
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|f)
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|width="50%" | {{:4.2 - Figure - An isosceles triangle with top angle v and sides 2, 3 and 3}}
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|}
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</div>{{#NAVCONTENT:Answer|Answer 4.2:2|Solution a |Solution 4.2:2a|Solution b |Solution 4.2:2b|Solution c |Solution 4.2:2c|Solution d |Solution 4.2:2d|Solution e |Solution 4.2:2e|Solution f |Solution 4.2:2f}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 40° och sidor 14 och x}}
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===Exercise 4.2:3===
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<div class="ovning">
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Determine
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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|Answer 4.2:3|Solution a |Solution 4.2:3a|Solution b |Solution 4.2:3b|Solution c |Solution 4.2:3c|Solution d |Solution 4.2:3d|Solution e |Solution 4.2:3e|Solution f |Solution 4.2:3f}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 20° och sidor 16 och x}}
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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|Answer 4.2:4|Solution a |Solution 4.2:4a|Solution b |Solution 4.2:4b|Solution c |Solution 4.2:4c|Solution d |Solution 4.2:4d|Solution e |Solution 4.2:4e|Solution f |Solution 4.2:4f}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 35° och sidor 11 och x}}
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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|Answer 4.2:5|Solution a |Solution 4.2:5a|Solution b |Solution 4.2:5b|Solution c |Solution 4.2:5c|Solution d |Solution 4.2:5d}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 32° och sidor x och 25}}
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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 - Figure - Two triangles with angles 45° and 60°, respectively, and height difference x}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 4.2:6|Solution |Solution 4.2:6}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln 50° och sidor x och 19}}
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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 - Figure - A river}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 4.2:7|Solution |Solution 4.2:7}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln v och sidor 2 och 5}}
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===Exercise 4.2:8===
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<div class="ovning">
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln v och sidor 70 och 110}}
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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 - Figure - Hanging rod}} </center>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 4.2:8|Solution |Solution 4.2:8}}
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln v och sidor 5 och 7}}
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===Exercise 4.2:9===
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<div class="ovning">
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{{:4.2 - Figur - Rätvinklig triangel med vinkeln v och sidor 3 och 5}}
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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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{{:4.2 - Figur - Rätvinklig triangel med vinklar v och 60° och sidan 5}}
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|
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|width="100%" | <center> {{:4.2 - Figure - A road from A to B via P and Q}} </center>
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{{:4.2 - Figur - Likbent triangel med toppvinkeln v och sidor 2, 3 och 3}}
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|}
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</div>{{#NAVCONTENT:Answer|Answer 4.2:9|Solution |Solution 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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Current revision

       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
Answer 4.2:1
Solution a
Solution 4.2:1a
Solution b
Solution 4.2:1b
Solution c
Solution 4.2:1c
Solution d
Solution 4.2:1d
Solution e
Solution 4.2:1e
Solution f
Solution 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
Answer 4.2:2
Solution a
Solution 4.2:2a
Solution b
Solution 4.2:2b
Solution c
Solution 4.2:2c
Solution d
Solution 4.2:2d
Solution e
Solution 4.2:2e
Solution f
Solution 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
Answer 4.2:3
Solution a
Solution 4.2:3a
Solution b
Solution 4.2:3b
Solution c
Solution 4.2:3c
Solution d
Solution 4.2:3d
Solution e
Solution 4.2:3e
Solution f
Solution 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
Answer 4.2:4
Solution a
Solution 4.2:4a
Solution b
Solution 4.2:4b
Solution c
Solution 4.2:4c
Solution d
Solution 4.2:4d
Solution e
Solution 4.2:4e
Solution f
Solution 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
Answer 4.2:5
Solution a
Solution 4.2:5a
Solution b
Solution 4.2:5b
Solution c
Solution 4.2:5c
Solution d
Solution 4.2:5d

Exercise 4.2:6

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

[Image]

Answer
Answer 4.2:6
Solution
Solution 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
Answer 4.2:7
Solution
Solution 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
Answer 4.2:8
Solution
Solution 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
Answer 4.2:9
Solution
Solution 4.2:9
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