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4.3 Exercises

From Förberedande kurs i matematik 1

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{{Ej vald flik|[[4.3 Trigonometriska samband|Theory]]}}
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{{Not selected tab|[[4.3 Trigonometric relations|Theory]]}}
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{{Vald flik|[[4.3 Övningar|Exercises]]}}
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{{Selected tab|[[4.3 Exercises|Exercises]]}}
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|width="33%" | <math>\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}</math>
|width="33%" | <math>\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}</math>
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</div>{{#NAVCONTENT:Answer|Answer 4.3:1|Solution a |Lösning 4.3:1a|Solution b |Lösning 4.3:1b|Solution c |Lösning 4.3:1c}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:1|Solution a |Solution 4.3:1a|Solution b |Solution 4.3:1b|Solution c |Solution 4.3:1c}}
===Exercise 4.3:2===
===Exercise 4.3:2===
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|width="50%" | <math>\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}</math>
|width="50%" | <math>\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}</math>
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</div>{{#NAVCONTENT:Answer|Answer 4.3:2|Solution a |Lösning 4.3:2a|Solution b |Lösning 4.3:2b}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:2|Solution a |Solution 4.3:2a|Solution b |Solution 4.3:2b}}
===Exercise 4.3:3===
===Exercise 4.3:3===
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|width="50%" | <math>\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}</math>
|width="50%" | <math>\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}</math>
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</div>{{#NAVCONTENT:Answer|Answer 4.3:3|Solution a |Lösning 4.3:3a|Solution b |Lösning 4.3:3b|Solution c |Lösning 4.3:3c|Solution d |Lösning 4.3:3d|Solution e |Lösning 4.3:3e|Solution f |Lösning 4.3:3f}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:3|Solution a |Solution 4.3:3a|Solution b |Solution 4.3:3b|Solution c |Solution 4.3:3c|Solution d |Solution 4.3:3d|Solution e |Solution 4.3:3e|Solution f |Solution 4.3:3f}}
===Exercise 4.3:4===
===Exercise 4.3:4===
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|width="50%" | <math>\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}</math>
|width="50%" | <math>\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}</math>
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</div>{{#NAVCONTENT:Answer|Answer 4.3:4|Solution a |Lösning 4.3:4a|Solution b |Lösning 4.3:4b|Solution c |Lösning 4.3:4c|Solution d |Lösning 4.3:4d|Solution e |Lösning 4.3:4e|Solution f |Lösning 4.3:4f}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:4|Solution a |Solution 4.3:4a|Solution b |Solution 4.3:4b|Solution c |Solution 4.3:4c|Solution d |Solution 4.3:4d|Solution e |Solution 4.3:4e|Solution f |Solution 4.3:4f}}
===Exercise 4.3:5===
===Exercise 4.3:5===
<div class="ovning">
<div class="ovning">
Determine <math>\,\cos{v}\,</math> and <math>\,\tan{v}\,</math>, where <math>\,v\,</math> is an acute angle in a triangle such that <math>\,\sin{v}=\displaystyle \frac{5}{7}\,</math>.
Determine <math>\,\cos{v}\,</math> and <math>\,\tan{v}\,</math>, where <math>\,v\,</math> is an acute angle in a triangle such that <math>\,\sin{v}=\displaystyle \frac{5}{7}\,</math>.
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</div>{{#NAVCONTENT:Answer|Answer 4.3:5|Solution |Lösning 4.3:5}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:5|Solution |Solution 4.3:5}}
===Exercise 4.3:6===
===Exercise 4.3:6===
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|width="100%" | Determine <math>\ \sin{v}\ </math> and <math>\ \cos{v}\ </math> if <math>\ \tan{v}=3\ </math> and <math>\ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,</math>.
|width="100%" | Determine <math>\ \sin{v}\ </math> and <math>\ \cos{v}\ </math> if <math>\ \tan{v}=3\ </math> and <math>\ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,</math>.
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</div>{{#NAVCONTENT:Answer|Answer 4.3:6|Solution a |Lösning 4.3:6a|Solution b |Lösning 4.3:6b|Solution c |Lösning 4.3:6c}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:6|Solution a |Solution 4.3:6a|Solution b |Solution 4.3:6b|Solution c |Solution 4.3:6c}}
===Exercise 4.3:7===
===Exercise 4.3:7===
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|width="100%" | <math>\cos{x}=\displaystyle \frac{2}{5}\,</math>, <math>\ \cos{y}=\displaystyle \frac{3}{5}\ </math> and <math>\,x\,</math>, <math>\,y\,</math> are angles in the first quadrant.
|width="100%" | <math>\cos{x}=\displaystyle \frac{2}{5}\,</math>, <math>\ \cos{y}=\displaystyle \frac{3}{5}\ </math> and <math>\,x\,</math>, <math>\,y\,</math> are angles in the first quadrant.
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</div>{{#NAVCONTENT:Answer|Answer 4.3:7|Solution a |Lösning 4.3:7a|Solution b |Lösning 4.3:7b}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:7|Solution a |Solution 4.3:7a|Solution b |Solution 4.3:7b}}
===Exercise 4.3:8===
===Exercise 4.3:8===
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|width="100%" | <math>\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v</math>
|width="100%" | <math>\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v</math>
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</div>{{#NAVCONTENT:Solution a |Lösning 4.3:8a|Solution b |Lösning 4.3:8b|Solution c |Lösning 4.3:8c|Solution d |Lösning 4.3:8d}}
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</div>{{#NAVCONTENT:Solution a |Solution 4.3:8a|Solution b |Solution 4.3:8b|Solution c |Solution 4.3:8c|Solution d |Solution 4.3:8d}}
===Exercise 4.3:9===
===Exercise 4.3:9===
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|width="100%" |(Hint: use the formula for double angles on <math>\,\sin 160^\circ\,</math>.)
|width="100%" |(Hint: use the formula for double angles on <math>\,\sin 160^\circ\,</math>.)
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</div>{{#NAVCONTENT:Solution |Lösning 4.3:9}}
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</div>{{#NAVCONTENT:Solution |Solution 4.3:9}}

Current revision

       Theory          Exercises      

Exercise 4.3:1

Determine the angles v between 2 and 2 which satisfy

a) cosv=cos5 b) sinv=sin7 c) tanv=tan72

Answer | Solution a | Solution b | Solution c

Exercise 4.3:2

Determine the angles v between 0 and which satisfy

a) cosv=cos23 b) cosv=cos57

Answer | Solution a | Solution b

Exercise 4.3:3

Suppose that −2v2 and that sinv=a. With the help of a express

a) sin(−v) b) sin(−v)
c) cosv d) sin2−v 
e) cos2+v  f) sin3+v 

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

Exercise 4.3:4

Suppose that 0v and that cosv=b. With the help of b express

a) sin2v b) sinv
c) sin2v d) cos2v
e) sinv+4  f) cosv−3 

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

Exercise 4.3:5

Determine cosv and tanv, where v is an acute angle in a triangle such that sinv=75.

Answer | Solution

Exercise 4.3:6

a) Determine  sinv  and  tanv  if  cosv=43  and  23v2.
b) Determine  cosv  and  tanv  if  sinv=310  and v lies in the second quadrant.
c) Determine  sinv  and  cosv  if  tanv=3  and  v23.

Answer | Solution a | Solution b | Solution c

Exercise 4.3:7

Determine  sin(x+y)  if

a) sinx=32, siny=31  and x, y are angles in the first quadrant.
b) cosx=52,  cosy=53  and x, y are angles in the first quadrant.

Answer | Solution a | Solution b

Exercise 4.3:8

Show the following trigonometric relations

a) tan2v=sin2v1−sin2v
b) 1cosv−tanv=cosv1+sinv
c) tan2u=sinu1+cosu
d) cos(u+v)cosucosv=1−tanutanv

Solution a | Solution b | Solution c | Solution d

Exercise 4.3:9

Show Feynman's equality
cos20cos40cos80=81.
(Hint: use the formula for double angles on sin160.)

Solution

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