Trigonometri merupakan salah satu cabang matematika yang mempelajari sudut dalam segitiga siku-siku (yang dijelaskan secara geometri). Identitas trigonometri merupakan salah satu fungsi trigonometri dimana rumus tersebut memiliki hasil yang sama bila diuji suatu nilai variabel . Identitas berikut ini sangatlah penting dan berguna dalam komputasi yang elusif.
Daftar ini menjelaskan dasar-dasar fungsi, invers fungsi, beserta nilai sudut istimewa pada fungsi trigonometri. Dan juga mengenai jumlah dan perkalian sudut. Mengenai daftar identitas fungsi invers juga dimasukkan ke dalam halaman ini. Terdapat bukti-bukti mengenai rumus-rumus di bawah. Meski begitu, halaman ini hanya menjelaskan bukti singkat pada rumus dan adapula yang tidak. Untuk melihat bukti, lihat Bukti identitas trigonometri . Berikut adalah daftar identitas trigonometri .
Segitiga siku-siku
A
B
C
{\displaystyle ABC}
A
C
=
b
{\displaystyle AC=b}
B
C
=
a
{\displaystyle BC=a}
sisi segitiga dan
A
B
=
c
{\displaystyle AB=c}
hipotenusa . Salah satu fungsi trigonometri paling umum, semenjak kita duduk di bangku sekolah menengah atas adalah fungsi trigonometri seperti sinus , kosinus , tangen , sekan , kosekan , dan kotangen . Secara geometri, keenam fungsi trigonometri tersebut dapat didefinisikan melalui sudut pada segitiga. Misalkan
A
B
C
{\displaystyle ABC}
segitiga siku-siku ,
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
hipotenusa atau sisi miring segitiga. Misalkan
A
{\displaystyle A}
sin
A
=
a
c
{\displaystyle \sin A={\frac {a}{c}}}
cos
A
=
b
c
{\displaystyle \cos A={\frac {b}{c}}}
tan
A
=
a
b
=
sin
A
cos
A
{\displaystyle \tan A={\frac {a}{b}}={\frac {\sin A}{\cos A}}}
cot
A
=
1
tan
A
=
cos
A
sin
A
=
b
a
{\displaystyle \cot A={\frac {1}{\tan A}}={\frac {\cos A}{\sin A}}={\frac {b}{a}}}
sec
A
=
1
cos
A
=
c
b
{\displaystyle \sec A={\frac {1}{\cos A}}={\frac {c}{b}}}
csc
A
=
1
sin
A
=
c
a
{\displaystyle \csc A={\frac {1}{\sin A}}={\frac {c}{a}}}
Keenam fungsi trigonometri di atas memiliki grafik, dengan ranah dan kisaran pada setiap dari mereka adalah berbeda, terutama periodenya. Berikut adalah daftar fungsi trigonometri yang ditabelkan, dengan periode, ranah, kisaran, beserta visualisasi grafik fungsi.
Fungsi
Periode
Ranah
Kisaran
Grafik
sinus
2
π
{\displaystyle 2\pi }
(
−
∞
,
∞
)
{\displaystyle (-\infty ,\infty )}
[
−
1
,
1
]
{\displaystyle [-1,1]}
kosinus
2
π
{\displaystyle 2\pi }
(
−
∞
,
∞
)
{\displaystyle (-\infty ,\infty )}
[
−
1
,
1
]
{\displaystyle [-1,1]}
tangen
π
{\displaystyle \pi }
x
≠
n
π
{\displaystyle x\neq n\pi }
(
−
∞
,
∞
)
{\displaystyle (-\infty ,\infty )}
sekan
2
π
{\displaystyle 2\pi }
x
≠
π
2
+
n
π
{\displaystyle x\neq {\frac {\pi }{2}}+n\pi }
(
−
∞
,
−
1
]
∪
[
1
,
∞
)
{\displaystyle (-\infty ,-1]\cup [1,\infty )}
kosekan
2
π
{\displaystyle 2\pi }
x
≠
π
2
+
n
π
{\displaystyle x\neq {\frac {\pi }{2}}+n\pi }
(
−
∞
,
−
1
]
∪
[
1
,
∞
)
{\displaystyle (-\infty ,-1]\cup [1,\infty )}
kotangen
π
{\displaystyle \pi }
x
≠
n
π
{\displaystyle x\neq n\pi }
(
−
∞
,
∞
)
{\displaystyle (-\infty ,\infty )}
Berikut adalah nilai sudut istimewa pada keenam fungsi trigonometri:
Sinus Kosinus Tangen Kotangen Sekan Kosekan
0°
0
{\displaystyle 0}
1
{\displaystyle 1}
0
{\displaystyle 0}
∞
{\displaystyle \infty }
1
{\displaystyle 1}
∞
{\displaystyle \infty }
8°
2
10
{\displaystyle {\frac {\sqrt {2}}{10}}}
7
2
10
{\displaystyle {\frac {7{\sqrt {2}}}{10}}}
1
7
{\displaystyle {\frac {1}{7}}}
7
{\displaystyle 7}
5
2
7
{\displaystyle {\frac {5{\sqrt {2}}}{7}}}
5
2
{\displaystyle 5{\sqrt {2}}}
15°
6
−
2
4
{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
6
+
2
4
{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
6
−
2
{\displaystyle {\sqrt {6}}-{\sqrt {2}}}
6
+
2
{\displaystyle {\sqrt {6}}+{\sqrt {2}}}
16°
7
25
{\displaystyle {\frac {7}{25}}}
24
25
{\displaystyle {\frac {24}{25}}}
7
24
{\displaystyle {\frac {7}{24}}}
24
7
{\displaystyle {\frac {24}{7}}}
25
24
{\displaystyle {\frac {25}{24}}}
25
7
{\displaystyle {\frac {25}{7}}}
18°
−
1
+
5
4
{\displaystyle {\frac {-1+{\sqrt {5}}}{4}}}
10
+
2
5
4
{\displaystyle {\frac {\sqrt {10+2{\sqrt {5}}}}{4}}}
(
−
5
+
3
5
)
10
+
2
5
20
{\displaystyle {\frac {(-5+3{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{20}}}
(
1
+
5
)
10
+
2
5
4
{\displaystyle {\frac {(1+{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{4}}}
(
5
−
5
)
10
+
2
5
10
{\displaystyle {\frac {(5-{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{10}}}
1
+
5
{\displaystyle 1+{\sqrt {5}}}
30°
1
2
{\displaystyle {\frac {1}{2}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
3
{\displaystyle {\sqrt {3}}}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
2
{\displaystyle 2}
36°
10
−
2
5
4
{\displaystyle {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
1
+
5
4
{\displaystyle {\frac {1+{\sqrt {5}}}{4}}}
(
−
1
+
5
)
10
−
2
5
4
{\displaystyle {\frac {(-1+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{4}}}
(
5
+
3
5
)
10
−
2
5
20
{\displaystyle {\frac {(5+3{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{20}}}
−
1
+
5
{\displaystyle -1+{\sqrt {5}}}
(
5
+
5
)
10
−
2
5
10
{\displaystyle {\frac {(5+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{10}}}
37°
3
5
{\displaystyle {\frac {3}{5}}}
4
5
{\displaystyle {\frac {4}{5}}}
3
4
{\displaystyle {\frac {3}{4}}}
4
3
{\displaystyle {\frac {4}{3}}}
5
4
{\displaystyle {\frac {5}{4}}}
5
3
{\displaystyle {\frac {5}{3}}}
45°
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
1
{\displaystyle 1}
1
{\displaystyle 1}
2
{\displaystyle {\sqrt {2}}}
2
{\displaystyle {\sqrt {2}}}
53°
4
5
{\displaystyle {\frac {4}{5}}}
3
5
{\displaystyle {\frac {3}{5}}}
4
3
{\displaystyle {\frac {4}{3}}}
3
4
{\displaystyle {\frac {3}{4}}}
5
3
{\displaystyle {\frac {5}{3}}}
5
4
{\displaystyle {\frac {5}{4}}}
54°
1
+
5
4
{\displaystyle {\frac {1+{\sqrt {5}}}{4}}}
10
−
2
5
4
{\displaystyle {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
(
5
+
3
5
)
10
−
2
5
20
{\displaystyle {\frac {(5+3{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{20}}}
(
−
1
+
5
)
10
−
2
5
4
{\displaystyle {\frac {(-1+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{4}}}
(
5
+
5
)
10
−
2
5
10
{\displaystyle {\frac {(5+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{10}}}
−
1
+
5
{\displaystyle -1+{\sqrt {5}}}
60°
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
1
2
{\displaystyle {\frac {1}{2}}}
3
{\displaystyle {\sqrt {3}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
2
{\displaystyle 2}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
72°
10
+
2
5
4
{\displaystyle {\frac {\sqrt {10+2{\sqrt {5}}}}{4}}}
−
1
+
5
4
{\displaystyle {\frac {-1+{\sqrt {5}}}{4}}}
(
1
+
5
)
10
+
2
5
4
{\displaystyle {\frac {(1+{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{4}}}
(
−
5
+
3
5
)
10
+
2
5
20
{\displaystyle {\frac {(-5+3{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{20}}}
1
+
5
{\displaystyle 1+{\sqrt {5}}}
(
5
−
5
)
10
+
2
5
10
{\displaystyle {\frac {(5-{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{10}}}
74°
24
25
{\displaystyle {\frac {24}{25}}}
7
25
{\displaystyle {\frac {7}{25}}}
24
7
{\displaystyle {\frac {24}{7}}}
7
24
{\displaystyle {\frac {7}{24}}}
25
7
{\displaystyle {\frac {25}{7}}}
25
24
{\displaystyle {\frac {25}{24}}}
75°
6
+
2
4
{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
6
−
2
4
{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
6
+
2
{\displaystyle {\sqrt {6}}+{\sqrt {2}}}
6
−
2
{\displaystyle {\sqrt {6}}-{\sqrt {2}}}
82°
7
2
10
{\displaystyle {\frac {7{\sqrt {2}}}{10}}}
2
10
{\displaystyle {\frac {\sqrt {2}}{10}}}
7
{\displaystyle 7}
1
7
{\displaystyle {\frac {1}{7}}}
5
2
{\displaystyle 5{\sqrt {2}}}
5
2
7
{\displaystyle {\frac {5{\sqrt {2}}}{7}}}
90°
1
{\displaystyle 1}
0
{\displaystyle 0}
∞
{\displaystyle \infty }
0
{\displaystyle 0}
∞
{\displaystyle \infty }
1
{\displaystyle 1}
Fungsi invers trigonometri merupakan fungsi yang merupakan kebalikan dari fungsi dasar trigonometri. Lazimnya, fungsi invers trigonometri biasanya dinotasikan dengan prefiks arc -.[ 1]
−
1
{\displaystyle ^{-1}}
[ nb 1]
Berikut adalah fungsi invers trigonometri, dengan ranah dan kisarannya, antara lain:
Nama fungsi
Simbol
Ranah
Citra/Kisaran
Fungsi
Ranah
Citra
sinus
sin
{\displaystyle \sin }
:
{\displaystyle :}
R
{\displaystyle \mathbb {R} }
→
{\displaystyle \to }
[
−
1
,
1
]
{\displaystyle [-1,1]}
arcsin
{\displaystyle \arcsin }
:
{\displaystyle :}
[
−
1
,
1
]
{\displaystyle [-1,1]}
→
{\displaystyle \to }
[
−
π
2
,
π
2
]
{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]}
kosinus
cos
{\displaystyle \cos }
:
{\displaystyle :}
R
{\displaystyle \mathbb {R} }
→
{\displaystyle \to }
[
−
1
,
1
]
{\displaystyle [-1,1]}
arccos
{\displaystyle \arccos }
:
{\displaystyle :}
[
−
1
,
1
]
{\displaystyle [-1,1]}
→
{\displaystyle \to }
[
0
,
π
]
{\displaystyle [0,\pi ]}
tangen
tan
{\displaystyle \tan }
:
{\displaystyle :}
π
Z
+
(
−
π
2
,
π
2
)
{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}
→
{\displaystyle \to }
R
{\displaystyle \mathbb {R} }
arctan
{\displaystyle \arctan }
:
{\displaystyle :}
R
{\displaystyle \mathbb {R} }
→
{\displaystyle \to }
(
−
π
2
,
π
2
)
{\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}
kotangen
cot
{\displaystyle \cot }
:
{\displaystyle :}
π
Z
+
(
0
,
π
)
{\displaystyle \pi \mathbb {Z} +(0,\pi )}
→
{\displaystyle \to }
R
{\displaystyle \mathbb {R} }
arccot
{\displaystyle \operatorname {arccot} }
:
{\displaystyle :}
R
{\displaystyle \mathbb {R} }
→
{\displaystyle \to }
(
0
,
π
)
{\displaystyle (0,\pi )}
sekan
sec
{\displaystyle \sec }
:
{\displaystyle :}
π
Z
+
(
−
π
2
,
π
2
)
{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}
→
{\displaystyle \to }
R
∖
(
−
1
,
1
)
=
(
−
∞
,
−
1
]
∪
[
1
,
∞
)
{\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )}
arcsec
{\displaystyle \operatorname {arcsec} }
:
{\displaystyle :}
R
∖
(
−
1
,
1
)
{\displaystyle \mathbb {R} \setminus (-1,1)}
→
{\displaystyle \to }
[
0
,
π
]
∖
{
π
2
}
{\displaystyle [\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}}
kosekan
csc
{\displaystyle \csc }
:
{\displaystyle :}
π
Z
+
(
0
,
π
)
{\displaystyle \pi \mathbb {Z} +(0,\pi )}
→
{\displaystyle \to }
R
∖
(
−
1
,
1
)
{\displaystyle \mathbb {R} \setminus (-1,1)}
arccsc
{\displaystyle \operatorname {arccsc} }
:
{\displaystyle :}
R
∖
(
−
1
,
1
)
{\displaystyle \mathbb {R} \setminus (-1,1)}
→
{\displaystyle \to }
[
−
π
2
,
π
2
]
∖
{
0
}
{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}}
Komposisi fungsi trigonometri dengan invers fungsinya sendiri akan sama dengan menuliskan suatu variabel. Dengan kata lain (tinjau
f
{\displaystyle f}
f
(
f
−
1
(
x
)
)
=
x
{\displaystyle f(f^{-1}(x))=x}
f
−
1
(
f
(
x
)
)
=
x
{\displaystyle f^{-1}(f(x))=x}
Hal yang serupa untuk fungsi trigonometri, berikut adalah fungsi yang memetakan fungsi inversnya sendiri:
sin
(
arcsin
(
x
)
)
=
x
{\displaystyle \sin(\arcsin(x))=x}
cos
(
arccos
(
x
)
)
=
x
{\displaystyle \cos(\arccos(x))=x}
tan
(
arctan
(
x
)
)
=
x
{\displaystyle \tan(\arctan(x))=x}
sec
(
arcsec
(
x
)
)
=
x
{\displaystyle \sec(\operatorname {arcsec}(x))=x}
csc
(
arccsc
(
x
)
)
=
x
{\displaystyle \csc(\operatorname {arccsc}(x))=x}
cot
(
arccot
(
x
)
)
=
x
{\displaystyle \cot(\operatorname {arccot}(x))=x}
Komposisi fungsi trigonometri dengan fungsi invers trigonometri lain
sunting
Komposisi fungsi invers untuk lebih lanjut dapat dilihat pada tabel di bawah ini.[ 2]
sin
(
arcsin
x
)
=
x
cos
(
arcsin
x
)
=
1
−
x
2
tan
(
arcsin
x
)
=
x
1
−
x
2
sin
(
arccos
x
)
=
1
−
x
2
cos
(
arccos
x
)
=
x
tan
(
arccos
x
)
=
1
−
x
2
x
sin
(
arctan
x
)
=
x
1
+
x
2
cos
(
arctan
x
)
=
1
1
+
x
2
tan
(
arctan
x
)
=
x
sin
(
arccsc
x
)
=
1
x
cos
(
arccsc
x
)
=
x
2
−
1
x
tan
(
arccsc
x
)
=
1
x
2
−
1
sin
(
arcsec
x
)
=
x
2
−
1
x
cos
(
arcsec
x
)
=
1
x
tan
(
arcsec
x
)
=
x
2
−
1
sin
(
arccot
x
)
=
1
1
+
x
2
cos
(
arccot
x
)
=
x
1
+
x
2
tan
(
arccot
x
)
=
1
x
{\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}}
Penyelesaian terhadap persamaan trigonometri
sunting
Berikut adalah penyelesaian persamaan trigonometri, dengan nilai
θ
{\displaystyle \theta }
x
{\displaystyle x}
Persamaan
⟺
{\displaystyle \iff }
Penyelesaian
untuk suatu
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
sin
θ
=
x
{\displaystyle \sin \theta =x}
θ
=
{\displaystyle \theta =\,}
(
−
1
)
k
{\displaystyle (-1)^{k}}
arcsin
(
x
)
{\displaystyle \arcsin(x)}
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
cos
θ
=
x
{\displaystyle \cos \theta =x}
θ
=
{\displaystyle \theta =\,}
±
{\displaystyle \pm \,}
arccos
(
x
)
{\displaystyle \arccos(x)}
+
{\displaystyle +}
2
{\displaystyle 2}
π
k
{\displaystyle \pi k}
tan
θ
=
x
{\displaystyle \tan \theta =x}
θ
=
{\displaystyle \theta =\,}
arctan
(
x
)
{\displaystyle \arctan(x)}
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
csc
θ
=
x
{\displaystyle \csc \theta =x}
θ
=
{\displaystyle \theta =\,}
(
−
1
)
k
{\displaystyle (-1)^{k}}
arccsc
(
x
)
{\displaystyle \operatorname {arccsc}(x)}
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
sec
θ
=
x
{\displaystyle \sec \theta =x}
θ
=
{\displaystyle \theta =\,}
±
{\displaystyle \pm \,}
arcsec
(
x
)
{\displaystyle \operatorname {arcsec}(x)}
+
{\displaystyle +}
2
{\displaystyle 2}
π
k
{\displaystyle \pi k}
cot
θ
=
x
{\displaystyle \cot \theta =x}
θ
=
{\displaystyle \theta =\,}
arccot
(
x
)
{\displaystyle \operatorname {arccot}(x)}
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
Berikut untuk persamaan dengan kedua ruas berupa fungsi trigonometri, tinjau sudut
θ
{\displaystyle \theta }
φ
{\displaystyle \varphi }
Persamaan
⟺
{\displaystyle \iff }
Penyelesaian
untuk suatu
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
Juga berlaku untuk persamaan
sin
θ
=
sin
φ
{\displaystyle \sin \theta =\sin \varphi }
θ
=
{\displaystyle \theta =\,}
(
−
1
)
k
{\displaystyle (-1)^{k}}
φ
{\displaystyle \varphi }
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
csc
θ
=
csc
φ
{\displaystyle \csc \theta =\csc \varphi }
cos
θ
=
cos
φ
{\displaystyle \cos \theta =\cos \varphi }
θ
=
{\displaystyle \theta =\,}
±
{\displaystyle \pm \,}
φ
{\displaystyle \varphi }
+
{\displaystyle +}
2
{\displaystyle 2}
π
k
{\displaystyle \pi k}
sec
θ
=
sec
φ
{\displaystyle \sec \theta =\sec \varphi }
tan
θ
=
tan
φ
{\displaystyle \tan \theta =\tan \varphi }
θ
=
{\displaystyle \theta =\,}
φ
{\displaystyle \varphi }
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
cot
θ
=
cot
φ
{\displaystyle \cot \theta =\cot \varphi }
−
sin
θ
=
sin
φ
{\displaystyle -\sin \theta =\sin \varphi }
θ
=
{\displaystyle \theta =\,}
(
−
1
)
k
+
1
{\displaystyle (-1)^{k+1}}
φ
{\displaystyle \varphi }
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
−
csc
θ
=
csc
φ
{\displaystyle -\csc \theta =\csc \varphi }
−
cos
θ
=
cos
φ
{\displaystyle -\cos \theta =\cos \varphi }
θ
=
{\displaystyle \theta =\,}
±
{\displaystyle \pm \,}
φ
{\displaystyle \varphi }
+
{\displaystyle +}
2
{\displaystyle 2}
π
k
+
π
{\displaystyle \pi k+\pi }
−
sec
θ
=
sec
φ
{\displaystyle -\sec \theta =\sec \varphi }
−
tan
θ
=
tan
φ
{\displaystyle -\tan \theta =\tan \varphi }
θ
=
{\displaystyle \theta =\,}
−
{\displaystyle -}
φ
{\displaystyle \varphi }
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
−
cot
θ
=
cot
φ
{\displaystyle -\cot \theta =\cot \varphi }
|
sin
θ
|
=
|
sin
φ
|
{\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|}
θ
=
{\displaystyle \theta =\,}
±
{\displaystyle \pm }
φ
{\displaystyle \varphi }
+
{\displaystyle +}
π
k
{\displaystyle \pi k}
|
cos
θ
|
=
|
cos
φ
|
|
tan
θ
|
=
|
tan
φ
|
|
csc
θ
|
=
|
csc
φ
|
|
sec
θ
|
=
|
csc
φ
|
|
cot
θ
|
=
|
csc
φ
|
{\displaystyle {\begin{aligned}\left|\cos \theta \right|&=\left|\cos \varphi \right|\\\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\csc \varphi \right|\\\left|\cot \theta \right|&=\left|\csc \varphi \right|\end{aligned}}}
Beberapa fungsi trigonometri lainnya
sunting
Beberapa fungsi trigonometri lainnya. Beberapa fungsi trigonometri antara lain: fungsi yang jarang digunakan seperti versin , coversin , vercosin , covercosin , haversin , havercosin , hacoversin , hacovercosin , exsec , dan excsc . Tabel di bawah menunjukkan fungsi trigonometri yang jarang digunakan beserta dengan grafiknya, antara lain sebagai berikut.
versin
(
θ
)
:=
2
sin
2
(
θ
2
)
=
1
−
cos
(
θ
)
{\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}
[ 3]
coversin
(
θ
)
:=
versin
(
π
2
−
θ
)
=
1
−
sin
(
θ
)
{\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}
[ 3]
vercosin
(
θ
)
:=
2
cos
2
(
θ
2
)
=
1
+
cos
(
θ
)
{\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}
[ 3]
covercosin
(
θ
)
:=
vercosin
(
π
2
−
θ
)
=
1
+
sin
(
θ
)
{\displaystyle {\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,}
[ 4]
haversin
(
θ
)
:=
versin
(
θ
)
2
=
sin
2
(
θ
2
)
=
1
−
cos
(
θ
)
2
{\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}=\sin ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}\,}
[ 5]
hacoversin
(
θ
)
:=
coversin
(
θ
)
2
=
1
−
sin
(
θ
)
2
{\displaystyle {\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,}
[ 6]
havercosin
(
θ
)
:=
vercosin
(
θ
)
2
=
cos
2
(
θ
2
)
=
1
+
cos
(
θ
)
2
{\displaystyle {\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}=\cos ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1+\cos(\theta )}{2}}\,}
[ 7]
hacovercosin
(
θ
)
:=
covercosin
(
θ
)
2
=
1
+
sin
(
θ
)
2
{\displaystyle {\textrm {hacovercosin}}(\theta ):={\frac {{\textrm {covercosin}}(\theta )}{2}}={\frac {1+\sin(\theta )}{2}}\,}
[ 8]
Selain fungsi yang jarang digunakan, terdapat fungsi trigonometri lainnya. Berikut di antaranya: tali busur disingkat crd , dan gd mengindikasikan fungsi Gudermann . Masing-masing dirumuskan sebagai berikut.
crd
θ
=
(
1
−
cos
θ
)
2
+
sin
2
θ
=
2
−
2
cos
θ
=
2
sin
(
θ
2
)
{\displaystyle \operatorname {crd} \ \theta ={\sqrt {(1-\cos \theta )^{2}+\sin ^{2}\theta }}={\sqrt {2-2\cos \theta }}=2\sin \left({\frac {\theta }{2}}\right)}
gd
x
=
∫
0
x
sech
t
d
t
{\displaystyle \operatorname {gd} x=\int _{0}^{x}\operatorname {sech} t\,\mathrm {d} t}
[ 9]
Untuk suatu fungsi trigonometri dasar beserta inversnya, dapat didefinisikan melalui eksponensiasi. Tabel berikut menunjukkannya.
Fungsi
Fungsi invers[ 14]
sin
θ
=
e
i
θ
−
e
−
i
θ
2
i
{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}
arcsin
θ
=
−
i
ln
(
i
θ
+
1
−
θ
2
)
{\displaystyle \arcsin \theta =-i\,\ln \left(i\theta +{\sqrt {1-\theta ^{2}}}\right)}
cos
θ
=
e
i
θ
+
e
−
i
θ
2
{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}
arccos
θ
=
−
i
ln
(
θ
+
θ
2
−
1
)
{\displaystyle \arccos \theta =-i\,\ln \left(\theta +\,{\sqrt {\theta ^{2}-1}}\right)}
tan
θ
=
−
i
e
i
θ
−
e
−
i
θ
e
i
θ
+
e
−
i
θ
{\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}}
arctan
θ
=
i
2
ln
(
i
+
θ
i
−
θ
)
{\displaystyle \arctan \theta ={\frac {i}{2}}\ln \left({\frac {i+\theta }{i-\theta }}\right)}
csc
θ
=
2
i
e
i
θ
−
e
−
i
θ
{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}
arccsc
θ
=
−
i
ln
(
i
θ
+
1
−
1
θ
2
)
{\displaystyle \operatorname {arccsc} \theta =-i\,\ln \left({\frac {i}{\theta }}+{\sqrt {1-{\frac {1}{\theta ^{2}}}}}\right)}
sec
θ
=
2
e
i
θ
+
e
−
i
θ
{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}
arcsec
θ
=
−
i
ln
(
1
θ
+
i
1
−
1
θ
2
)
{\displaystyle \operatorname {arcsec} \theta =-i\,\ln \left({\frac {1}{\theta }}+i{\sqrt {1-{\frac {1}{\theta ^{2}}}}}\right)}
cot
θ
=
i
e
i
θ
+
e
−
i
θ
e
i
θ
−
e
−
i
θ
{\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}}
arccot
θ
=
i
2
ln
(
θ
−
i
θ
+
i
)
{\displaystyle \operatorname {arccot} \theta ={\frac {i}{2}}\ln \left({\frac {\theta -i}{\theta +i}}\right)}
cis
θ
=
e
i
θ
{\displaystyle \operatorname {cis} \theta =e^{i\theta }}
arccis
θ
=
−
i
ln
θ
{\displaystyle \operatorname {arccis} \theta =-i\ln \theta }
Disini,
e
{\displaystyle e}
konstanta dengan nilai
2.718281845
…
{\displaystyle 2.718281845\dots }
i
{\displaystyle i}
bilangan imajiner , dan
cis
{\displaystyle \operatorname {cis} }
cis
θ
=
cos
θ
+
i
sin
θ
{\displaystyle \operatorname {cis} \theta =\cos \theta +i\sin \theta }
[ 15] [ 16] Pada tabel terakhir, baris awal dan kolom akhir, tepat di bawah kiri sel, rumus tersebut disebut juga sebagai rumus Euler .
Sudut rangkap merupakan sudut yang dimana suatu variabel yang sama ditambahkan oleh variabel tersendiri. Sudut rangkap dapat dibuktikan melalui sifat jumlah sudut. Sebagai contoh, ketika kita ingin mencari
sin
2
x
{\displaystyle \sin 2x}
sin
2
θ
=
sin
(
θ
+
θ
)
=
sin
θ
cos
θ
+
sin
θ
cos
θ
=
2
sin
θ
cos
θ
{\displaystyle {\begin{aligned}\sin 2\theta &=\sin({\color {red}{\theta }}+{\color {green}{\theta }})\\&=\sin {\color {red}{\theta }}\cos {\color {green}{\theta }}+\sin {\color {green}{\theta }}\cos {\color {red}{\theta }}\\&=2\sin \theta \cos \theta \end{aligned}}}
Rumus jumlah sudut tersebut juga kita pakai untuk mencari sudut rangkap tiga. Andaikan kita diminta untuk mencari
sin
3
x
{\displaystyle \sin 3x}
sin
3
θ
=
sin
(
2
θ
+
θ
)
=
sin
2
θ
cos
θ
+
sin
θ
cos
2
θ
=
2
sin
θ
cos
2
θ
+
sin
θ
(
1
−
2
sin
2
θ
)
=
2
sin
θ
(
1
−
sin
2
θ
)
+
sin
θ
−
2
sin
3
θ
=
2
sin
θ
−
2
sin
3
θ
+
sin
θ
−
2
sin
3
θ
=
3
sin
θ
−
4
sin
3
θ
{\displaystyle {\begin{aligned}\sin 3\theta &=\sin(2\theta +\theta )\\&=\sin 2\theta \cos \theta +\sin \theta \cos 2\theta \\&=2\sin \theta \cos ^{2}\theta +\sin \theta (1-2\sin ^{2}\theta )\\&=2\sin \theta (1-\sin ^{2}\theta )+\sin \theta -2\sin ^{3}\theta \\&=2\sin \theta -2\sin ^{3}\theta +\sin \theta -2\sin ^{3}\theta \\&=3\sin \theta -4\sin ^{3}\theta \end{aligned}}}
Dengan cara yang serupa, kita dapat mencari rumus untuk fungsi trigonometri sudut rangkap lainnya, seperti kosinus, tangen, kotangen, sekan, serta dengan kosekan.
Kita telah memperoleh rumus sudut rangkap dua dan sudut rangkap tiga (pada kotak di samping), maka kita beralih ke sudut
n
{\displaystyle n}
n
=
1
,
2
,
3
…
{\displaystyle n=1,2,3\dots }
n
{\displaystyle n}
n
{\displaystyle n}
n
=
2
{\displaystyle n=2}
Tanpa basa-basi, berikut adalah rumus sudut
n
{\displaystyle n}
[ 27]
sin
(
n
x
)
=
∑
k
=
0
n
(
n
k
)
cos
k
x
sin
n
−
k
x
sin
(
π
2
(
n
−
k
)
)
{\displaystyle \sin(nx)=\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}x\sin ^{n-k}x\sin \left({\frac {\pi }{2}}(n-k)\right)}
cos
(
n
x
)
=
∑
k
=
0
n
(
n
k
)
cos
k
x
sin
n
−
k
x
cos
(
π
2
(
n
−
k
)
)
{\displaystyle \cos(nx)=\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}x\sin ^{n-k}x\cos \left({\frac {\pi }{2}}(n-k)\right)}
Metode Chebyshev adalah algoritme rekursif yang mencari rumus sudut
n
{\displaystyle n}
(
n
−
1
)
{\displaystyle (n-1)}
(
n
−
2
)
{\displaystyle (n-2)}
[ 28]
sin
(
n
x
)
=
2
cos
x
sin
(
(
n
−
1
)
x
)
−
sin
(
(
n
−
2
)
x
)
cos
(
n
x
)
=
2
cos
x
cos
(
(
n
−
1
)
x
)
−
cos
(
(
n
−
2
)
x
)
tan
(
n
x
)
=
tan
(
(
n
−
1
)
x
)
+
tan
x
1
−
tan
(
(
n
−
1
)
x
)
tan
x
{\displaystyle {\begin{aligned}\sin(nx)&=2\cos x\sin((n-1)x)-\sin((n-2)x)\\\cos(nx)&=2\cos x\cos((n-1)x)-\cos((n-2)x)\\\tan(nx)&={\frac {\tan((n-1)x)+\tan x}{1-\tan((n-1)x)\tan x}}\end{aligned}}}
Berikut adalah sudut setengah rangkap, antara lain[ 29] [ 30]
sin
θ
2
=
±
1
−
cos
θ
2
cos
θ
2
=
±
1
+
cos
θ
2
tan
θ
2
=
csc
θ
−
cot
θ
=
±
1
−
cos
θ
1
+
cos
θ
=
sin
θ
1
+
cos
θ
=
1
−
cos
θ
sin
θ
=
−
1
±
1
+
tan
2
θ
tan
θ
=
tan
θ
1
+
sec
θ
cot
θ
2
=
csc
θ
+
cot
θ
=
±
1
+
cos
θ
1
−
cos
θ
=
sin
θ
1
−
cos
θ
=
1
+
cos
θ
sin
θ
{\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&=\pm {\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\pm {\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta =\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}={\frac {-1\pm {\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}={\frac {\tan \theta }{1+\sec {\theta }}}\\[3pt]\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta =\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}={\frac {\sin \theta }{1-\cos \theta }}={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}
Masih terdapat rumus-rumus lainnya berkaitan dengan sudut setengah rangkap. Berikut di antaranya:
tan
η
±
θ
2
=
sin
η
±
sin
θ
cos
η
+
cos
θ
tan
(
θ
2
+
π
4
)
=
sec
θ
+
tan
θ
1
−
sin
θ
1
+
sin
θ
=
|
1
−
tan
θ
2
|
|
1
+
tan
θ
2
|
{\displaystyle {\begin{aligned}\tan {\frac {\eta \pm \theta }{2}}&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}\\[3pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\end{aligned}}}
Penjumlahan dan perkalian fungsi trigonometri
sunting
Contoh limit fungsi trigonometri yang paling sering digunakan adalah
lim
x
→
0
sin
x
x
=
1
{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}
Kita dapat membuktikan contoh pertama dengan menggunakan satuan lingkaran dan teorema apit . Terdapat contoh limit yang juga paling sering dipakai,
lim
x
→
0
1
−
cos
x
x
=
0
{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}
Limit tersebut dapat dibuktikan melalui fungsi trigonometri tangen rangkap setengah. Untuk limit fungsi trigonometri lainnya, berikut adalah limit fungsi trigonometri beserta dengan pembuktiannya.
lim
x
→
0
sin
a
x
b
x
=
lim
x
→
0
a
x
sin
b
x
=
lim
x
→
0
sin
a
x
sin
b
x
=
a
b
{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{bx}}=\lim _{x\to 0}{\frac {ax}{\sin bx}}=\lim _{x\to 0}{\frac {\sin ax}{\sin bx}}={\frac {a}{b}}}
[ nb 2] Untuk suatu fungsi trigonometri, terdapat turunan dan antiturunan . Walakin, halaman ini hanya menjelaskan turunan dan antiturunan terhadap fungsi trigonometri yang bersifat dasar beserta fungsi inversnya. Untuk mengenai antiturunan fungsi trigonometri lainnya, lihat Daftar integral dari fungsi trigonometri dan Daftar integral dari fungsi invers trigonometri . Tabel berikut ini menunjukkan turunan dan antiturunan, antara lain:
Turunan
Integral
d
d
x
sin
x
=
cos
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sin x=\cos x}
∫
sin
x
d
x
=
−
cos
x
+
C
{\displaystyle \int \sin x\,\mathrm {d} x=-\cos x+C}
d
d
x
cos
x
=
−
sin
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cos x=-\sin x}
∫
cos
d
x
=
sin
x
+
C
{\displaystyle \int \cos \,\mathrm {d} x=\sin x+C}
d
d
x
tan
x
=
sec
2
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\tan x=\sec ^{2}x}
∫
tan
x
d
x
=
−
ln
|
cos
x
|
+
C
{\displaystyle \int \tan x\,\mathrm {d} x=-\ln \left|\cos x\right|+C}
d
d
x
csc
x
=
−
cot
x
csc
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\csc x=-\cot x\csc x}
∫
csc
x
d
x
=
−
ln
|
csc
x
+
cot
x
|
+
C
{\displaystyle \int \csc x\,\mathrm {d} x=-\ln \left|\csc x+\cot x\right|+C}
d
d
x
sec
x
=
tan
x
sec
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sec x=\tan x\sec x}
∫
sec
x
d
x
=
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec x\,\mathrm {d} x=\ln \left|\sec x+\tan x\right|+C}
d
d
x
cot
x
=
−
csc
2
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cot x=-\csc ^{2}x}
∫
cot
x
d
x
=
ln
|
sin
x
|
+
C
{\displaystyle \int \cot x\,\mathrm {d} x=\ln \left|\sin x\right|+C}
sunting
^ Misalnya, invers fungsi trigonometri sinus, dinotasikan
arcsin
(
…
)
{\displaystyle \arcsin(\dots )}
sin
−
1
(
…
)
{\displaystyle \sin ^{-1}(\dots )}
^ Sifat berikut juga memiliki beragam limit fungsi trigonometri yang sama dengan
a
b
{\displaystyle {\frac {a}{b}}}
lim
x
→
0
tan
a
x
b
x
{\displaystyle \lim _{x\to 0}{\frac {\tan ax}{bx}}}
lim
x
→
0
a
x
tan
b
x
{\displaystyle \lim _{x\to 0}{\frac {ax}{\tan bx}}}
lim
x
→
0
tan
a
x
tan
b
x
{\displaystyle \lim _{x\to 0}{\frac {\tan ax}{\tan bx}}}
lim
x
→
0
sin
a
x
tan
b
x
=
lim
x
→
0
tan
a
x
sin
b
x
{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{\tan bx}}=\lim _{x\to 0}{\frac {\tan ax}{\sin bx}}}
^ Hall, Arthur Graham; Frink, Fred Goodrich ([c1909]). Plane trigonometry […] α = arcsin m : It is frequently read "arc-sine m " or "anti-sine m ," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1 m , is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […] ^ Abramowitz & Stegun 1972 , hlm. 73, 4.3.45^ a b c "Abramowitz and Stegun. Page 78" . personal.math.ubc.ca . Diakses tanggal 2021-12-05 . ^ Weisstein, Eric W. "Covercosine" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-05 . ^ Weisstein, Eric W. "Haversine" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-05 . ^ Weisstein, Eric W. "Hacoversine" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-05 . ^ Weisstein, Eric W. "Havercosine" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-07 . ^ Weisstein, Eric W. "Hacovercosine" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-05 . ^ Weisstein, Eric W. "Gudermannian" . MathWorld ^ "Trigonometric Identities | Boundless Algebra" . courses.lumenlearning.com . Diakses tanggal 2021-11-26 . ^ Abramowitz and Stegun, hlm. 73, 4.3.45
^ Abramowitz and Stegun, hlm. 72, 4.3.13–15
^ Abramowitz and Stegun, hlm. 72, 4.3.7–9
^ Abramowitz and Stegun, hlm. 80, 4.4.26–31
^ Weisstein, Eric W. "Cis" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-11-29 . ^ "Mathwords: Cis" . www.mathwords.com . Diakses tanggal 2021-11-29 . ^ "7.2: Sum and Difference Identities" . Mathematics LibreTexts (dalam bahasa Inggris). 2015-10-31. Diakses tanggal 2021-12-02 . ^ Abramowitz and Stegun, hlm. 72, 4.3.16
^ a b c d (Inggris) Weisstein, Eric W. "Trigonometric Addition Formulas" . MathWorld ^ Abramowitz and Stegun, hlm. 72, 4.3.17
^ Abramowitz and Stegun, hlm. 72, 4.3.18
^ a b "Angle Sum and Difference Identities" . www.milefoot.com . Diakses tanggal 2019-10-12 . ^ Abramowitz and Stegun, hlm. 72, 4.3.19
^ Abramowitz and Stegun, hlm. 80, 4.4.32
^ Abramowitz and Stegun, hlm. 80, 4.4.33
^ Abramowitz and Stegun, hlm. 80, 4.4.34
^ Weisstein, Eric W. "Multiple-Angle Formulas" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-11-29 . ^ "Cosine, Sine and Tangent of Multiple Angles (Recursive Formula)" . trans4mind.com . Diakses tanggal 2021-12-02 . ^ Abramowitz and Stegun, hlm. 72, 4.3.20–22
^ (Inggris) Weisstein, Eric W. "Half-Angle Formulas" . MathWorld ^ Abramowitz and Stegun, hlm. 72, 4.3.31–33
^ Abramowitz and Stegun, hlm. 72, 4.3.34–39
^ Weisstein, Eric W. "Tangent" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-07 . ^ Weisstein, Eric W. "Cosecant" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-07 . ^ Weisstein, Eric W. "Secant" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-07 . ^ Weisstein, Eric W. "Cotangent" . mathworld.wolfram.com (dalam bahasa Inggris). Diakses tanggal 2021-12-07 .
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![{\displaystyle [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751)
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![{\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&=\pm {\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\pm {\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta =\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}={\frac {-1\pm {\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}={\frac {\tan \theta }{1+\sec {\theta }}}\\[3pt]\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta =\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}={\frac {\sin \theta }{1-\cos \theta }}={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc1f00b70f6eddf80b91c17e5f785480bc325a8)
![{\displaystyle {\begin{aligned}\tan {\frac {\eta \pm \theta }{2}}&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}\\[3pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/639270245e163f1afa8682ac8aa12086db43dce1)
![{\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{dimana }}S=\{1,-1\}^{n}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/219f5560a5a505f3d3c48c68d987097bbb84cb2f)
