Definition von Sinus Kosinus und Tangens [ Bearbeiten ]
sin
α
=
a
b
=
3
5
cos
α
=
c
b
=
4
5
tan
α
=
a
c
=
3
4
{\displaystyle \sin \alpha ={\frac {a}{b}}={\frac {3}{5}}\quad \cos \alpha ={\frac {c}{b}}={\frac {4}{5}}\quad \tan \alpha ={\frac {a}{c}}={\frac {3}{4}}}
sin
γ
=
c
f
=
8
17
cos
γ
=
g
f
=
15
17
tan
γ
=
c
g
=
8
15
{\displaystyle \sin \gamma ={\frac {c}{f}}={\frac {8}{17}}\quad \cos \gamma ={\frac {g}{f}}={\frac {15}{17}}\quad \tan \gamma ={\frac {c}{g}}={\frac {8}{15}}}
sin
θ
=
y
r
=
1
2
cos
θ
=
x
r
=
3
2
tan
θ
=
y
x
=
3
3
{\displaystyle \sin \theta ={\frac {y}{r}}={\frac {1}{2}}\quad \cos \theta ={\frac {x}{r}}={\frac {\sqrt {3}}{2}}\quad \tan \theta ={\frac {y}{x}}={\frac {\sqrt {3}}{3}}}
sin
ε
=
e
m
=
1
2
cos
ε
=
f
m
=
3
2
tan
ε
=
e
f
=
3
3
{\displaystyle \sin \varepsilon ={\frac {e}{m}}={\frac {1}{2}}\quad \cos \varepsilon ={\frac {f}{m}}={\frac {\sqrt {3}}{2}}\quad \tan \varepsilon ={\frac {e}{f}}={\frac {\sqrt {3}}{3}}}
cos
=
12
13
{\displaystyle \cos ={\frac {12}{13}}}
θ
≈
22
,
62
∘
{\displaystyle \theta \approx 22{,}62^{\circ }}
23
,
4
c
m
{\displaystyle 23{,}4\ cm}
ω
=
arccos
(
c
b
)
≈
57
,
28
∘
{\displaystyle \omega =\arccos \left({\tfrac {c}{b}}\right)\approx 57{,}28^{\circ }}
c
a
.
66
,
5
∘
{\displaystyle ca.\ 66{,}5^{\circ }}
cos
=
20
29
{\displaystyle \cos ={\frac {20}{29}}}
θ
≈
46
,
40
∘
{\displaystyle \theta \approx 46{,}40^{\circ }}
17
20
21
c
m
≈
17
,
95
c
m
{\displaystyle 17{\frac {20}{21}}\,cm\approx 17{,}95\ cm}
θ
=
arctan
(
y
x
)
≈
28
,
39
∘
{\displaystyle \theta =\arctan \left({\tfrac {y}{x}}\right)\approx 28{,}39^{\circ }}
c
a
.
1
,
72
∘
{\displaystyle ca.\ 1{,}72^{\circ }}
cos
=
72
97
{\displaystyle \cos ={\frac {72}{97}}}
θ
≈
42
,
08
∘
{\displaystyle \theta \approx 42{,}08^{\circ }}
19
,
4
c
m
{\displaystyle 19{,}4\ cm}
ϵ
=
arctan
(
e
f
)
≈
39
,
04
∘
{\displaystyle \epsilon =\arctan \left({\tfrac {e}{f}}\right)\approx 39{,}04^{\circ }}
c
a
.
59
,
53
∘
{\displaystyle ca.\ 59{,}53^{\circ }}
cos
=
36
85
{\displaystyle \cos ={\frac {36}{85}}}
θ
≈
64
,
94
∘
{\displaystyle \theta \approx 64{,}94^{\circ }}
85
7
≈
12
,
14
c
m
{\displaystyle {\tfrac {85}{7}}\approx 12{,}14\ cm}
ϕ
=
arccos
(
e
m
)
≈
35
,
82
∘
{\displaystyle \phi =\arccos \left({\tfrac {e}{m}}\right)\approx 35{,}82^{\circ }}
c
a
.
11
,
31
∘
{\displaystyle ca.\ 11{,}31^{\circ }}
Pythagoras Satz in Trigonometrie Abstrakt [ Bearbeiten ] a, b: Katheten, c: Hypotenuse.
a
2
+
b
2
=
c
2
→
c
2
sin
2
α
+
c
2
cos
2
α
=
c
2
→
{\displaystyle a^{2}+b^{2}=c^{2}\ \rightarrow \ c^{2}\ \sin ^{2}\alpha +c^{2}\cos ^{2}\alpha =c^{2}\ \rightarrow \ }
c
2
(
sin
2
α
+
cos
2
α
)
=
c
2
→
sin
2
α
+
cos
2
α
=
1
{\displaystyle c^{2}\ (\sin ^{2}\alpha +\cos ^{2}\alpha )=c^{2}\ \rightarrow \ \sin ^{2}\alpha +\cos ^{2}\alpha =1}
tan
x
=
Gegenkathete
Ankathete
=
e
r
w
e
i
t
e
r
n
Gegenkathete
Hypotenuse
Ankathete
Hypotenuse
=
sin
x
cos
x
{\displaystyle \textstyle \tan x={\frac {\text{Gegenkathete}}{\text{Ankathete}}}\quad {\overset {erweitern}{=}}\quad {\frac {\quad {\frac {\text{Gegenkathete}}{\text{Hypotenuse}}}\quad }{\quad {\frac {\text{Ankathete}}{\text{Hypotenuse}}}\quad }}={\frac {\sin x}{\cos x}}}
tan
2
x
=
sin
2
x
cos
2
x
=
1
−
cos
2
x
cos
2
x
=
1
cos
2
x
−
cos
2
x
cos
2
x
=
1
cos
2
x
−
1
{\displaystyle \textstyle \tan ^{2}x={\frac {\sin ^{2}x}{\cos ^{2}x}}={\frac {1-\cos ^{2}x}{\cos ^{2}x}}={\frac {1}{\cos ^{2}x}}-{\frac {\cos ^{2}x}{\cos ^{2}x}}={\frac {1}{\cos ^{2}x}}-1}
hier klicken Pythagoras Satz in Trigonometrie Konkret [ Bearbeiten ]
tan
α
=
20
21
sin
α
=
c
o
s
β
=
20
29
cos
α
=
21
29
{\displaystyle \tan \alpha ={\tfrac {20}{21}}\quad \sin \alpha =cos\beta ={\tfrac {20}{29}}\quad \cos \alpha ={\tfrac {21}{29}}}
α
≈
43
,
60
∘
β
≈
46
,
40
∘
{\displaystyle \alpha \approx 43{,}60^{\circ }\quad \beta \approx 46{,}40^{\circ }}
tan
α
=
26
69
sin
α
=
c
o
s
β
=
69
269
cos
α
=
260
269
{\displaystyle \tan \alpha ={\tfrac {26}{69}}\quad \sin \alpha =cos\beta ={\tfrac {69}{269}}\quad \cos \alpha ={\tfrac {260}{269}}}
α
≈
14
,
86
∘
β
≈
75
,
14
∘
{\displaystyle \alpha \approx 14{,}86^{\circ }\quad \beta \approx 75{,}14^{\circ }}
tan
α
=
2
3
sin
α
=
c
o
s
β
=
2
13
cos
α
=
3
13
{\displaystyle \tan \alpha ={\tfrac {2}{3}}\quad \sin \alpha =cos\beta ={\tfrac {2}{\sqrt {13}}}\quad \cos \alpha ={\tfrac {3}{\sqrt {13}}}}
α
≈
33
,
69
∘
β
≈
56
,
31
∘
{\displaystyle \alpha \approx 33{,}69^{\circ }\quad \beta \approx 56{,}31^{\circ }}
tan
α
=
11
,
9
12
sin
α
=
c
o
s
β
=
119
169
cos
α
=
120
169
{\displaystyle \tan \alpha ={\tfrac {11{,}9}{12}}\quad \sin \alpha =cos\beta ={\tfrac {119}{169}}\quad \cos \alpha ={\tfrac {120}{169}}}
α
≈
44
,
76
∘
β
≈
45
,
24
∘
{\displaystyle \alpha \approx 44{,}76^{\circ }\quad \beta \approx 45{,}24^{\circ }}
Einheitskreis und trigonometrische Funktionen [ Bearbeiten ] i) 17,46°+n·360° oder 162,54°+n·360°
i) −17,46°+n·360° oder 197,46°+n·360°
i) 53,13°+n·360° oder 126,87°+n·360°
i) 315°+n·360° oder 225°+n·360° A)
540
∘
,
{\displaystyle \ 540^{\circ },\quad }
100
∘
,
{\displaystyle \ 100^{\circ },\quad }
c
a
.
11
,
5
∘
,
{\displaystyle \ ca.\ 11{,}5^{\circ },\quad }
c
a
.
10
,
1
∘
,
{\displaystyle \ ca.\ 10{,}1^{\circ },\quad }
c
a
.
20626
∘
{\displaystyle \ ca.\ 20626^{\circ }\quad }
A)
c
a
.
5
,
24
r
a
d
,
{\displaystyle \ ca.\ 5{,}24\ rad,\quad }
c
a
.
0,003
5
r
a
d
,
{\displaystyle \ ca.\ 0{,}0035\ rad,\quad }
c
a
.
0,164
r
a
d
,
{\displaystyle \ ca.\ 0{,}164\ rad,\quad }
0,105
r
a
d
,
{\displaystyle \ 0{,}105\ rad,\quad }
3
π
4
r
a
d
{\displaystyle \textstyle \ {\frac {3\pi }{4}}\ rad\quad }
A) 4.Q
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
A)
900
∘
,
{\displaystyle \ 900^{\circ },\quad }
c
a
.
231
∘
,
{\displaystyle \ ca.\ 231^{\circ },\quad }
c
a
.
34
,
4
∘
,
{\displaystyle \ ca.\ 34{,}4^{\circ },\quad }
c
a
.
14
,
6
∘
,
{\displaystyle \ ca.\ 14{,}6^{\circ },\quad }
c
a
.
15470
∘
{\displaystyle \ ca.\ 15470^{\circ }\quad }
A)
c
a
.
3
,
49
r
a
d
,
{\displaystyle \ ca.\ 3{,}49\ rad,\quad }
c
a
.
0,008
7
r
a
d
,
{\displaystyle \ ca.\ 0{,}0087\ rad,\quad }
c
a
.
0,274
r
a
d
,
{\displaystyle \ ca.\ 0{,}274\ rad,\quad }
0,361
r
a
d
,
{\displaystyle \ 0{,}361\ rad,\quad }
19
π
10
r
a
d
{\displaystyle \textstyle \ {\frac {19\pi }{10}}\ rad\quad }
A) 3.Q
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
A)
720
∘
,
{\displaystyle \ 720^{\circ },\quad }
252
∘
,
{\displaystyle \ 252^{\circ },\quad }
c
a
.
28
,
6
∘
,
{\displaystyle \ ca.\ 28{,}6^{\circ },\quad }
c
a
.
11
,
4
∘
,
{\displaystyle \ ca.\ 11{,}4^{\circ },\quad }
c
a
.
18048
∘
{\displaystyle \ ca.\ 18048^{\circ }\quad }
A)
1
,
3
8
˙
π
r
a
d
,
{\displaystyle \ 1{,}3{\dot {8}}\pi \ rad,\quad }
c
a
.
0,007
0
r
a
d
,
{\displaystyle \ ca.\ 0{,}0070\ rad,\quad }
c
a
.
0,246
r
a
d
,
{\displaystyle \ ca.\ 0{,}246\ rad,\quad }
0,361
r
a
d
,
{\displaystyle \ 0{,}361\ rad,\quad }
c
a
.
3
,
80
r
a
d
{\displaystyle \textstyle \ ca.\ 3{,}80\ rad\quad }
A) 2.Q
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
A)
1260
∘
,
{\displaystyle \ 1260^{\circ },\quad }
210
∘
,
{\displaystyle \ 210^{\circ },\quad }
c
a
.
85
,
9
∘
,
{\displaystyle \ ca.\ 85{,}9^{\circ },\quad }
c
a
.
14
,
2
∘
,
{\displaystyle \ ca.\ 14{,}2^{\circ },\quad }
c
a
.
24100
∘
{\displaystyle \ ca.\ 24100^{\circ }\quad }
A)
π
12
r
a
d
,
{\displaystyle \ {\tfrac {\pi }{12}}\ rad,\quad }
π
225
r
a
d
,
{\displaystyle \ \ {\tfrac {\pi }{225}}rad,\quad }
7
π
2
360
r
a
d
,
{\displaystyle \ {\tfrac {7\pi ^{2}}{360}}\ rad,\quad }
43
252
r
a
d
,
{\displaystyle \ {\tfrac {43}{252}}\ rad,\quad }
14
π
9
r
a
d
{\displaystyle {\tfrac {14\pi }{9}}\ rad\quad }
A) 1.Q
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
{\displaystyle \quad }
Sinus + in 1. & 2. Q., Kosinus + in 1. & 4. Q., Tangens + in 1. & 3. Q.
sin
x
=
0
⇔
x
=
n
⋅
π
r
a
d
=
n
⋅
180
∘
,
n
∈
Z
{\displaystyle \sin x=0\Leftrightarrow x=n\cdot \pi \ rad=n\cdot \ 180^{\circ },\ n\in \mathbb {Z} }
sin
x
=
1
⇔
x
=
(
2
n
⋅
π
+
π
2
)
r
a
d
=
n
⋅
360
∘
+
90
∘
,
n
∈
Z
{\displaystyle \sin x=1\Leftrightarrow x=\left(2\,n\cdot \pi +\textstyle {\frac {\pi }{2}}\right)\ rad=n\cdot \ 360^{\circ }+90^{\circ },\ n\in \mathbb {Z} }
sin
x
=
−
1
⇔
x
=
(
2
n
⋅
π
−
π
2
)
r
a
d
=
n
⋅
360
∘
−
90
∘
,
n
∈
Z
{\displaystyle \sin x=-1\Leftrightarrow x=\left(2\,n\cdot \pi -\textstyle {\frac {\pi }{2}}\right)\ rad=n\cdot \ 360^{\circ }-90^{\circ },\ n\in \mathbb {Z} }
Sinus + in 1. & 2. Q., Kosinus + in 1. & 4. Q., Tangens + in 1. & 3. Q.
sin
x
=
c
o
s
x
>
0
⇔
x
=
(
2
n
⋅
π
+
π
4
)
r
a
d
=
n
⋅
360
∘
+
45
∘
,
n
∈
Z
{\displaystyle \sin x=cosx>0\Leftrightarrow x=\left(2\,n\cdot \pi +\textstyle {\frac {\pi }{4}}\right)\ rad=n\cdot \ 360^{\circ }+45^{\circ },\ n\in \mathbb {Z} }
cos
x
=
1
2
⇔
x
=
(
2
n
⋅
π
±
π
3
)
r
a
d
=
n
⋅
360
∘
±
60
∘
,
n
∈
Z
{\displaystyle \textstyle \cos x={\frac {1}{2}}\Leftrightarrow x=\left(2\,n\cdot \pi \pm {\frac {\pi }{3}}\right)\ rad=n\cdot \ 360^{\circ }\pm 60^{\circ },\ n\in \mathbb {Z} }
Sinus + in 1. & 2. Q., Kosinus + in 1. & 4. Q., Tangens + in 1. & 3. Q.
cos
x
=
1
⇔
x
=
2
n
⋅
π
r
a
d
=
n
⋅
360
∘
,
n
∈
Z
{\displaystyle \cos x=1\Leftrightarrow x=2\,n\cdot \pi \ rad=n\cdot \ 360^{\circ },\ n\in \mathbb {Z} }
cos
x
=
0
⇔
x
=
(
n
⋅
π
+
π
2
)
r
a
d
=
n
⋅
180
∘
+
90
∘
,
n
∈
Z
{\displaystyle \cos x=0\Leftrightarrow x=\left(n\cdot \pi +\textstyle {\frac {\pi }{2}}\right)\ rad=n\cdot \ 180^{\circ }+90^{\circ },\ n\in \mathbb {Z} }
cos
x
=
−
1
⇔
x
=
(
2
n
+
1
)
⋅
π
r
a
d
=
(
2
n
+
1
)
⋅
180
∘
,
n
∈
Z
{\displaystyle \cos x=-1\Leftrightarrow x=(2\,n+1)\cdot \pi \ rad=(2\,n+1)\cdot \ 180^{\circ },\ n\in \mathbb {Z} }
Sinus + in 1. & 2. Q., Kosinus + in 1. & 4. Q., Tangens + in 1. & 3. Q.
cos
x
=
1
2
⇔
x
=
(
2
n
⋅
π
±
π
4
)
r
a
d
=
2
n
⋅
360
∘
±
45
∘
,
n
∈
Z
{\displaystyle \cos x={\tfrac {1}{2}}\Leftrightarrow x=\left(2\ n\cdot \pi \pm {\tfrac {\pi }{4}}\right)\ rad=2\ n\cdot \ 360^{\circ }\pm 45^{\circ },\ n\in \mathbb {Z} }
cos
x
=
−
1
2
⇔
x
=
(
2
n
+
1
±
π
4
)
⋅
π
r
a
d
=
(
2
n
+
1
±
1
4
)
⋅
180
∘
,
n
∈
Z
{\displaystyle \cos x=-{\tfrac {1}{2}}\Leftrightarrow x=(2\,n+1\pm {\tfrac {\pi }{4}})\cdot \pi \ rad=(2\,n+1\pm {\tfrac {1}{4}})\cdot \ 180^{\circ },\ n\in \mathbb {Z} }
Parameter im Diagramm der Sinusfunktion [ Bearbeiten ] blau 1,1; rot 0,6 rot, ? Blau sin, Orange cos, grün tan
A
0
=
2
,
4
,
ω
=
3
,
c
=
0
,
4
,
ϕ
=
0
{\displaystyle A0=2{,}4,\ \omega =3,\ c=0{,}4,\ \phi =0}
blau −1,4; rot 2,4 rot,
±
π
{\displaystyle \pm \pi }
Blau sin, Orange cos, grün tan
A
0
=
0
,
6
,
ω
=
2
,
c
=
0
,
ϕ
=
−
π
4
{\displaystyle A0=0{,}6,\ \omega =2,\ c=0,\ \phi =-{\tfrac {\pi }{4}}}
blau 3; rot 1 keine, beide
±
π
{\displaystyle \pm \pi }
Blau sin, Orange cos, grün tan
A
0
=
1
,
ω
=
1
,
c
=
1
,
ϕ
=
−
π
4
{\displaystyle A0=1,\ \omega =1,\ c=1,\ \phi =-{\tfrac {\pi }{4}}}
blau 0; rot 1,6 keine, blau
±
π
{\displaystyle \pm \pi }
−
3
π
4
{\displaystyle -{\tfrac {3\pi }{4}}}
Blau sin, Orange cos, grün tan
A
0
=
2
,
ω
=
1
,
c
=
−
0
,
8
,
ϕ
=
π
2
{\displaystyle A0=2,\ \omega =1,\ c=-0{,}8,\ \phi ={\tfrac {\pi }{2}}}
blau 2,6; rot 1,2 rot, blau
±
π
{\displaystyle \pm \pi }
Blau sin, Orange cos, grün tan
A
0
=
3
,
4
,
ω
=
1
2
,
c
=
0
,
ϕ
=<
m
a
t
h
>
±
π
{\displaystyle A0=3{,}4,\ \omega ={\tfrac {1}{2}},\ c=0,\ \phi =<math>\pm \pi }
blau 0,6; rot 3,2 keine, blau
±
π
{\displaystyle \pm \pi }
−
π
4
{\displaystyle -{\tfrac {\pi }{4}}}
Blau sin, Orange cos, grün tan
A
0
=
0
,
6
,
ω
=
1
4
,
c
=
−
1
,
4
,
ϕ
=<
m
a
t
h
>
±
π
{\displaystyle A0=0{,}6,\ \omega ={\tfrac {1}{4}},\ c=-1{,}4,\ \phi =<math>\pm \pi }
Direkte Anwendung des Sinus und des Kosinussatzes [ Bearbeiten ]
125
,
93
∘
{\displaystyle 125{,}93^{\circ }}
d
2
≈
4
,
3
c
m
{\displaystyle d_{2}\approx 4{,}3\ cm}
a
≈
3
,
94
c
m
{\displaystyle a\approx 3{,}94\ cm}
d
2
≈
3
,
53
c
m
{\displaystyle d_{2}\approx 3{,}53\ cm}
d
≈
3
,
95
c
m
{\displaystyle d\approx 3{,}95\ cm}
d
2
≈
9
,
24
c
m
{\displaystyle d_{2}\approx 9{,}24\ cm}
h
≈
7
,
56
m
,
d
≈
2
,
47
m
{\displaystyle h\approx 7{,}56\ m,\quad d\approx 2{,}47\ m}
d
≈
1037
m
{\displaystyle d\approx 1037\ m}
h
≈
13
,
54
,
d
≈
2
,
84
{\displaystyle h\approx 13{,}54,\quad d\approx 2{,}84}
d
≈
2826
m
{\displaystyle d\approx 2826\ m}
h
≈
11
,
55
,
d
≈
3
,
86
{\displaystyle h\approx 11{,}55,\quad d\approx 3{,}86}
d
≈
1719
m
{\displaystyle d\approx 1719\ m}
_pictogram.svg){kind=small}
