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4.3 Exercises

From Förberedande kurs i matematik 1

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{{Mall:Ej vald flik|[[4.3 Trigonometriska samband|Teori]]}}
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{{Not selected tab|[[4.3 Trigonometric relations|Theory]]}}
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{{Mall:Vald flik|[[4.3 Övningar|Övningar]]}}
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{{Selected tab|[[4.3 Exercises|Exercises]]}}
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===Övning 4.3:1===
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===Exercise 4.3:1===
<div class="ovning">
<div class="ovning">
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Bestäm de vinklar <math>\,v\,</math> mellan <math>\,\displaystyle \frac{\pi}{2}\,</math> och <math>\,2\pi\,</math> som uppfyller
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Determine the angles <math>\,v\,</math> between <math>\,\displaystyle \frac{\pi}{2}\,</math> and <math>\,2\pi\,</math> which satisfy
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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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:Svar|Svar 4.3:1|Lösning a |Lösning 4.3:1a|Lösning b |Lösning 4.3:1b|Lösning 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}}
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===Övning 4.3:2===
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===Exercise 4.3:2===
<div class="ovning">
<div class="ovning">
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Bestäm de vinklar <math>\,v\,</math> mellan 0 och <math>\,\pi\,</math> som uppfyller
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Determine the angles <math>\,v\,</math> between 0 and <math>\,\pi\,</math> which satisfy
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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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:Svar|Svar 4.3:2|Lösning a |Lösning 4.3:2a|Lösning 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}}
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===Övning 4.3:3===
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===Exercise 4.3:3===
<div class="ovning">
<div class="ovning">
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Antag att <math>\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\,</math> och att <math>\,\sin{v} = a\,</math>. Uttryck med hjälp av <math>\,a</math>
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Suppose that <math>\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\,</math> and that <math>\,\sin{v} = a\,</math>. With the help of <math>\,a</math> express
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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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:Svar|Svar 4.3:3|Lösning a |Lösning 4.3:3a|Lösning b |Lösning 4.3:3b|Lösning c |Lösning 4.3:3c|Lösning d |Lösning 4.3:3d|Lösning e |Lösning 4.3:3e|Lösning 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}}
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===Övning 4.3:4===
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===Exercise 4.3:4===
<div class="ovning">
<div class="ovning">
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Antag att <math>\,0 \leq v \leq \pi\,</math> och att <math>\,\cos{v}=b\,</math>. Uttryck med hjälp av <math>\,b</math>
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Suppose that <math>\,0 \leq v \leq \pi\,</math> and that <math>\,\cos{v}=b\,</math>. With the help of <math>\,b</math> express
{| width="100%" cellspacing="10px"
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|a)
|a)
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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:Svar|Svar 4.3:4|Lösning a |Lösning 4.3:4a|Lösning b |Lösning 4.3:4b|Lösning c |Lösning 4.3:4c|Lösning d |Lösning 4.3:4d|Lösning e |Lösning 4.3:4e|Lösning 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}}
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===Övning 4.3:5===
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===Exercise 4.3:5===
<div class="ovning">
<div class="ovning">
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För en spetsig vinkel <math>\,v\,</math> i en triangel gäller att <math>\,\sin{v}=\displaystyle \frac{5}{7}\,</math>. Bestäm <math>\,\cos{v}\,</math> och <math>\,\tan{v}\,</math>.
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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:Svar|Svar 4.3:5|Lösning |Lösning 4.3:5}}
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</div>{{#NAVCONTENT:Answer|Answer 4.3:5|Solution |Solution 4.3:5}}
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===Övning 4.3:6===
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===Exercise 4.3:6===
<div class="ovning">
<div class="ovning">
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | Bestäm <math>\ \sin{v}\ </math> och <math>\ \tan{v}\ </math> om <math>\ \cos{v}=\displaystyle \frac{3}{4}\ </math> och <math>\ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,</math>.
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|width="100%" | Determine <math>\ \sin{v}\ </math> and <math>\ \tan{v}\ </math> if <math>\ \cos{v}=\displaystyle \frac{3}{4}\ </math> and <math>\ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,</math>.
|-
|-
|b)
|b)
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|width="100%" | Bestäm <math>\ \cos{v}\ </math> och <math>\ \tan{v}\ </math> om <math>\ \sin{v}=\displaystyle \frac{3}{10}\ </math> och <math>\,v\,</math> ligger i den andra kvadranten.
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|width="100%" | Determine <math>\ \cos{v}\ </math> and <math>\ \tan{v}\ </math> if <math>\ \sin{v}=\displaystyle \frac{3}{10}\ </math> and <math>\,v\,</math> lies in the second quadrant.
|-
|-
|c)
|c)
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|width="100%" | Bestäm <math>\ \sin{v}\ </math> och <math>\ \cos{v}\ </math> om <math>\ \tan{v}=3\ </math> och <math>\ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,</math>.
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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>.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:6|Lösning a |Lösning 4.3:6a|Lösning b |Lösning 4.3:6b|Lösning 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}}
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===Övning 4.3:7===
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===Exercise 4.3:7===
<div class="ovning">
<div class="ovning">
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Bestäm <math>\ \sin{(x+y)}\ </math> om
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Determine <math>\ \sin{(x+y)}\ </math> if
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | <math>\sin{x}=\displaystyle \frac{2}{3}\,</math>,<math>\ \sin{y}=\displaystyle \frac{1}{3}\ </math> och <math>\,x\,$, $\,y\,</math> är vinklar i första kvadranten..
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|width="100%" | <math>\sin{x}=\displaystyle \frac{2}{3}\,</math>,<math>\ \sin{y}=\displaystyle \frac{1}{3}\ </math> and <math>\,x\,</math>, <math> \,y\,</math> are angles in the first quadrant.
|-
|-
|b)
|b)
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|width="100%" | <math>\cos{x}=\displaystyle \frac{2}{5}\,</math>, <math>\ \cos{y}=\displaystyle \frac{3}{5}\ </math> och <math>\,x\,</math>, <math>\,y\,</math> är vinklar i första kvadranten.
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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.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 4.3:7|Lösning a |Lösning 4.3:7a|Lösning 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}}
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===Exercise 4.3:8===
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<div class="ovning">
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Show the following trigonometric relations
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{| width="100%" cellspacing="10px"
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|a)
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|width="100%" | <math>\tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}</math>
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|-
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|b)
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|width="100%" | <math>\displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v}</math>
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|-
 +
|c)
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|width="100%" | <math>\tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}</math>
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|-
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|d)
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|width="100%" | <math>\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v</math>
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|}
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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}}
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===Exercise 4.3:9===
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<div class="ovning">
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{| width="100%" cellspacing="10px"
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|
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|width="100%" | Show Feynman's equality
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|-
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|
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|width="100%" |<center> <math>\cos 20^\circ \cdot \cos 40^\circ \cdot \cos 80^\circ = \displaystyle\frac{1}{8}\,\mbox{.}</math> </center>
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|-
 +
|
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|width="100%" |(Hint: use the formula for double angles on <math>\,\sin 160^\circ\,</math>.)
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|}
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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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