Övningsuppgifter på trigonometri Uppgift 1: Beräkna längden av hypotenusan för följande trianglar
a) b)
c) d)
e) f)
g) h)
Uppgift 2: Beräkna höjden i följande trianglar
a) b)
c) d)
α=77,0˚
45 cm 11,0 cm
α=59˚
α=21
10 cm ˚ 22 cm
α=29
˚
α=69˚
13,5 cm
9,0 cm α=66,0˚
α=27˚
17 cm 12 cm
α=36˚
α=69˚
7,0 cm 6,0 cm
α=66˚
α=27˚
9,0 cm 8,0 cm
α=36˚
e) f)
g) h)
Uppgift 3: Beräkna längden av basen i följande trianglar
a) b)
c) d)
e) f)
g) h)
α=69˚
13,5 cm
9,0 cm α=66˚
α=27˚
17 cm 12 cm
α=36˚
α=62˚
13,5 cm 9,0 cm
α=61˚
α=22˚
17 cm 12 cm
α=33˚
α=69˚
6,0 cm α=66˚ 4,0 cm
α=27˚
5,0 cm 7,0 cm
α=36˚
Uppgift 4: Beräkna vinklarna α och β i följande trianglar
a) b)
c) d)
e) f)
g) h)
13,5 cm 9,0 cm
α
β α
β 7,2 cm
8,0 cm
17 cm 12 cm
α β
α β
13,5 cm
4,3 cm
6,0 cm 4,0 cm
9,0 cm
6,8 cm α
β
α β
α
5,0 cm 7,0 cm
11,0 cm β
13,0 cm
α β
Facit Uppgift 1:
a) b)
cosα HYP =
NK =cosα
HYP NK NK HYP
α =
cos NK HYP
α = cos cm
HYP 10
27 cos
9 ≈
= HYP 9,9cm
36 cos
8 ≈
=
c) d)
cosα HYP =
NK =cosα
HYP NK NK HYP
α =
cos NK HYP
α = cos cm
HYP 11
21 cos
10 ≈
= HYP 25cm
29 cos
22 ≈
=
e) f)
sinα HYP =
MK =sinα
HYP MK MK HYP
α =
sin MK HYP
α = sin cm
HYP 7,7 66 sin
7 ≈
= HYP 6,4cm
69 sin
6 ≈
=
g) h)
sinα HYP =
MK =sinα
HYP MK MK HYP
α =
sin MK HYP
α = sin cm
HYP 52
59 sin
45 ≈
= HYP 11,3cm
77 sin
11 ≈
= Uppgift 2: Beräkna höjden i följande trianglar
a) b)
sinα HYP =
MK =sinα
HYP MK sinα
⋅
= HYP
MK MK = HYP⋅sinα
cm
MK =17⋅sin27≈7,7 MK=12⋅sin36≈7,1cm
c) d)
cosα HYP =
NK =cosα
HYP NK cosα
⋅
= HYP
NK NK = HYP⋅cosα
cm
NK =13,5⋅cos66≈5,49 NK =9⋅cos69≈3,2cm
e) f)
tanα NK =
MK =tanα
NK MK tanα
⋅
MK = NK MK = NK⋅tanα cm
NK =17⋅tan22≈6,9 NK =12⋅tan33≈7,8cm
g) h) tanα
NK =
MK =tanα
NK MK
tanα NK = MK
tanα NK = MK cm
NK 7,5 61 tan
5 ,
13 ≈
= NK 4,8cm
62 tan
9 ≈
= Uppgift 3:
a) b)
tanα NK =
MK =tanα
NK MK
tanα NK = MK
tanα NK = MK cm
NK 41
27 tan
5 ≈
= NK 9,6cm
36 tan
7 ≈
=
c) d)
tanα NK =
MK =tanα
NK MK tanα
⋅
MK = NK MK = NK⋅tanα cm
NK =6⋅tan66≈13 NK =4⋅tan69≈10cm
e) f)
cosα HYP =
NK =cosα
HYP NK cosα
⋅
= HYP
NK NK = HYP⋅cosα
cm
NK =17⋅cos27≈15 NK =12⋅cos36≈9,7cm
g) h)
sinα HYP =
MK =sinα
HYP MK sinα
⋅
= HYP
MK MK = HYP⋅sinα
cm
MK =13,5⋅sin66≈12 MK=9⋅sin69≈8,4cm Uppgift 4:
a) a)
tanα NK =
MK =tanβ
NK MK tanα
11
5 = tanβ
5 11=
°
≈
= ) 24
11 arctan(5
α = )≈66°
5 arctan(11 β
b) b)
tanα NK =
MK =tanβ
NK MK tanα
13
7 = tanβ
7 13 =
°
≈
= ) 28
13 arctan(7
α = )≈62°
7 arctan(13 β
c) c)
tanα NK =
MK =tanβ
NK MK tanα
9
6 = tanβ
6 9 =
°
≈
= ) 34
9 arctan(6
α = )≈56°
6 arctan(9 β
d) d)
sinα HYP =
MK =cosβ
HYP NK sinα
8 , 6
4 = cosβ
8 , 6
4 =
°
≈
= ) 36
8 , 6 arcsin( 4
α = )≈54°
8 , 6 arccos( 4 β
e) e)
sinβ HYP =
MK =cosα
HYP NK sinβ
17 5 ,
13 = cosα
17 5 , 13 =
°
≈
= ) 53
17 5 , arcsin(13
β = )≈37°
17 5 , arccos(13 α
f) f)
sinα HYP =
MK =cosβ
HYP NK sinα
12 3 ,
4 = cosβ
12 3 ,
4 =
°
≈
= ) 21
12 3 , arcsin(4
α = )≈69°
12 3 , arccos(4 β
g) g)
sinα HYP =
MK =cosβ
HYP NK sinα
5 , 13
2 ,
7 = cosβ
5 , 13
2 ,
7 =
°
≈
= ) 32
5 , 13
2 , arcsin(7
α = )≈58°
5 , 13
2 , arccos(7 β
h) h)
sinβ HYP =
MK =cosα
HYP NK sinβ
9
8 = cosα
9 8 =
°
≈
= ) 63
9 arcsin(8
β = )≈27°
9 arccos(8 α
