4.2 Übungen

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Exercise 4.2:1

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

a)

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b)

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c)

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d)

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e)

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f)

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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) 4.2 - Figure - A right-angled triangle with angle v and sides 2 and 5 b) 4.2 - Figure - A right-angled triangle with angle v and sides 70 and 110
c) 4.2 - Figure - A right-angled triangle with angle v and sides 5 and 7 d) 4.2 - Figure - A right-angled triangle with angle v and sides 3 and 5
e) 4.2 - Figure - A right-angled triangle with angles v and 60° and side 5 f) 4.2 - Figure - An isosceles triangle with top angle v and sides 2, 3 and 3
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\,.

4.2 - Figure - Two triangles with angles 45° and 60°, respectively, and height difference x
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?

4.2 - Figure - A river
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.

4.2 - Figure - Hanging rod
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.)

4.2 - Figure - A road from A to B via P and Q
Answer
Answer 4.2:9
Solution
Solution 4.2:9
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