Definisi ekponensial
Fungsi
Fungsi terbalik[1]
sin
θ
=
e
i
θ
−
e
−
i
θ
2
i
{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}
arcsin
x
=
−
i
ln
(
i
x
+
1
−
x
2
)
{\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)}
cos
θ
=
e
i
θ
+
e
−
i
θ
2
{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}
arccos
x
=
−
i
ln
(
x
+
x
2
−
1
)
{\displaystyle \arccos x=-i\,\ln \left(x+\,{\sqrt {x^{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
x
=
i
2
ln
(
i
+
x
i
−
x
)
{\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)}
csc
θ
=
2
i
e
i
θ
−
e
−
i
θ
{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}
arccsc
x
=
−
i
ln
(
i
x
+
1
−
1
x
2
)
{\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}
sec
θ
=
2
e
i
θ
+
e
−
i
θ
{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}
arcsec
x
=
−
i
ln
(
1
x
+
i
1
−
1
x
2
)
{\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{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
x
=
i
2
ln
(
x
−
i
x
+
i
)
{\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)}
cis
θ
=
e
i
θ
{\displaystyle \operatorname {cis} \theta =e^{i\theta }}
arccis
x
=
−
i
ln
x
{\displaystyle \operatorname {arccis} x=-i\ln x}
Indetitas untuk kasus α + β + γ = 180°
tan
α
+
tan
β
+
tan
γ
=
tan
α
⋅
tan
β
⋅
tan
γ
{\displaystyle \tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \cdot \tan \beta \cdot \tan \gamma \,}
cot
β
⋅
cot
γ
+
cot
γ
⋅
cot
α
+
cot
α
⋅
cot
β
=
1
{\displaystyle \cot \beta \cdot \cot \gamma +\cot \gamma \cdot \cot \alpha +\cot \alpha \cdot \cot \beta =1}
cot
α
2
+
cot
β
2
+
cot
γ
2
=
cot
α
2
⋅
cot
β
2
⋅
cot
γ
2
{\displaystyle \cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}=\cot {\frac {\alpha }{2}}\cdot \cot {\frac {\beta }{2}}\cdot \cot {\frac {\gamma }{2}}}
tan
β
2
tan
γ
2
+
tan
γ
2
tan
α
2
+
tan
α
2
tan
β
2
=
1
{\displaystyle \tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}+\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}=1}
sin
α
+
sin
β
+
sin
γ
=
4
cos
α
2
cos
β
2
cos
γ
2
{\displaystyle \sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}
−
sin
α
+
sin
β
+
sin
γ
=
4
cos
α
2
sin
β
2
sin
γ
2
{\displaystyle -\sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}
cos
α
+
cos
β
+
cos
γ
=
4
sin
α
2
sin
β
2
sin
γ
2
+
1
{\displaystyle \cos \alpha +\cos \beta +\cos \gamma =4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1}
−
cos
α
+
cos
β
+
cos
γ
=
4
sin
α
2
cos
β
2
cos
γ
2
−
1
{\displaystyle -\cos \alpha +\cos \beta +\cos \gamma =4\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}-1}
sin
(
2
α
)
+
sin
(
2
β
)
+
sin
(
2
γ
)
=
4
sin
α
sin
β
sin
γ
{\displaystyle \sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )=4\sin \alpha \sin \beta \sin \gamma \,}
−
sin
(
2
α
)
+
sin
(
2
β
)
+
sin
(
2
γ
)
=
4
sin
α
cos
β
cos
γ
{\displaystyle -\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )=4\sin \alpha \cos \beta \cos \gamma \,}
cos
(
2
α
)
+
cos
(
2
β
)
+
cos
(
2
γ
)
=
−
4
cos
α
cos
β
cos
γ
−
1
{\displaystyle \cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )=-4\cos \alpha \cos \beta \cos \gamma -1\,}
−
cos
(
2
α
)
+
cos
(
2
β
)
+
cos
(
2
γ
)
=
−
4
cos
α
sin
β
sin
γ
+
1
{\displaystyle -\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )=-4\cos \alpha \sin \beta \sin \gamma +1\,}
sin
2
α
+
sin
2
β
+
sin
2
γ
=
2
cos
α
cos
β
cos
γ
+
2
{\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2\cos \alpha \cos \beta \cos \gamma +2\,}
−
sin
2
α
+
sin
2
β
+
sin
2
γ
=
2
cos
α
sin
β
sin
γ
{\displaystyle -\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2\cos \alpha \sin \beta \sin \gamma \,}
cos
2
α
+
cos
2
β
+
cos
2
γ
=
−
2
cos
α
cos
β
cos
γ
+
1
{\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =-2\cos \alpha \cos \beta \cos \gamma +1\,}
−
cos
2
α
+
cos
2
β
+
cos
2
γ
=
−
2
cos
α
sin
β
sin
γ
+
1
{\displaystyle -\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =-2\cos \alpha \sin \beta \sin \gamma +1\,}
−
sin
2
(
2
α
)
+
sin
2
(
2
β
)
+
sin
2
(
2
γ
)
=
−
2
cos
(
2
α
)
sin
(
2
β
)
sin
(
2
γ
)
{\displaystyle -\sin ^{2}(2\alpha )+\sin ^{2}(2\beta )+\sin ^{2}(2\gamma )=-2\cos(2\alpha )\sin(2\beta )\sin(2\gamma )}
−
cos
2
(
2
α
)
+
cos
2
(
2
β
)
+
cos
2
(
2
γ
)
=
2
cos
(
2
α
)
sin
(
2
β
)
sin
(
2
γ
)
+
1
{\displaystyle -\cos ^{2}(2\alpha )+\cos ^{2}(2\beta )+\cos ^{2}(2\gamma )=2\cos(2\alpha )\,\sin(2\beta )\,\sin(2\gamma )+1}
sin
2
(
α
2
)
+
sin
2
(
β
2
)
+
sin
2
(
γ
2
)
+
2
sin
(
α
2
)
sin
(
β
2
)
sin
(
γ
2
)
=
1
{\displaystyle \sin ^{2}\left({\frac {\alpha }{2}}\right)+\sin ^{2}\left({\frac {\beta }{2}}\right)+\sin ^{2}\left({\frac {\gamma }{2}}\right)+2\sin \left({\frac {\alpha }{2}}\right)\,\sin \left({\frac {\beta }{2}}\right)\,\sin \left({\frac {\gamma }{2}}\right)=1}
Komposisi pada fungsi trigonometri
cos
(
t
sin
x
)
=
J
0
(
t
)
+
2
∑
k
=
1
∞
J
2
k
(
t
)
cos
(
2
k
x
)
{\displaystyle \cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)}
sin
(
t
sin
x
)
=
2
∑
k
=
0
∞
J
2
k
+
1
(
t
)
sin
(
(
2
k
+
1
)
x
)
{\displaystyle \sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}}
cos
(
t
cos
x
)
=
J
0
(
t
)
+
2
∑
k
=
1
∞
(
−
1
)
k
J
2
k
(
t
)
cos
(
2
k
x
)
{\displaystyle \cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)}
sin
(
t
cos
x
)
=
2
∑
k
=
0
∞
(
−
1
)
k
J
2
k
+
1
(
t
)
cos
(
(
2
k
+
1
)
x
)
{\displaystyle \sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}}
Rumus pada Produk tak terhingga[2] [3]
sin
x
=
x
∏
n
=
1
∞
(
1
−
x
2
π
2
n
2
)
sinh
x
=
x
∏
n
=
1
∞
(
1
+
x
2
π
2
n
2
)
cos
x
=
∏
n
=
1
∞
(
1
−
x
2
π
2
(
n
−
1
2
)
2
)
cosh
x
=
∏
n
=
1
∞
(
1
+
x
2
π
2
(
n
−
1
2
)
2
)
{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\\\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)\end{aligned}}\ \,{\begin{aligned}\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)^{2}}}\right)\\\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)^{2}}}\right)\end{aligned}}}
Fungsi trigonometri terbalik
arcsin
x
+
arccos
x
=
π
2
arctan
x
+
arccot
x
=
π
2
arctan
x
+
arctan
1
x
=
{
π
2
,
if
x
>
0
−
π
2
,
if
x
<
0
{\displaystyle {\begin{aligned}\arcsin x+\arccos x&={\dfrac {\pi }{2}}\\\arctan x+\operatorname {arccot} x&={\dfrac {\pi }{2}}\\\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\dfrac {\pi }{2}},&{\text{if }}x>0\\-{\dfrac {\pi }{2}},&{\text{if }}x<0\end{cases}}\end{aligned}}}
arctan
1
x
=
arctan
1
x
+
y
+
arctan
y
x
2
+
x
y
+
1
{\displaystyle \arctan {\frac {1}{x}}=\arctan {\frac {1}{x+y}}+\arctan {\frac {y}{x^{2}+xy+1}}}
[4] Compositions of trig and inverse trig functions
sin
(
arccos
x
)
=
1
−
x
2
tan
(
arcsin
x
)
=
x
1
−
x
2
sin
(
arctan
x
)
=
x
1
+
x
2
tan
(
arccos
x
)
=
1
−
x
2
x
cos
(
arctan
x
)
=
1
1
+
x
2
cot
(
arcsin
x
)
=
1
−
x
2
x
cos
(
arcsin
x
)
=
1
−
x
2
cot
(
arccos
x
)
=
x
1
−
x
2
{\displaystyle {\begin{aligned}\sin(\arccos x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cot(\arcsin x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\cot(\arccos x)&={\frac {x}{\sqrt {1-x^{2}}}}\end{aligned}}}
Jumlah lain dari fungsi trigonometri[5]
sin
φ
+
sin
(
φ
+
α
)
+
sin
(
φ
+
2
α
)
+
⋯
⋯
+
sin
(
φ
+
n
α
)
=
sin
(
n
+
1
)
α
2
⋅
sin
(
φ
+
n
α
2
)
sin
α
2
dan
cos
φ
+
cos
(
φ
+
α
)
+
cos
(
φ
+
2
α
)
+
⋯
⋯
+
cos
(
φ
+
n
α
)
=
sin
(
n
+
1
)
α
2
⋅
cos
(
φ
+
n
α
2
)
sin
α
2
.
{\displaystyle {\begin{aligned}&\sin \varphi +\sin(\varphi +\alpha )+\sin(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\sin(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \sin \left(\varphi +{\frac {n\alpha }{2}}\right)}{\sin {\frac {\alpha }{2}}}}\quad {\text{dan}}\\[10pt]&\cos \varphi +\cos(\varphi +\alpha )+\cos(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\cos(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \cos \left(\varphi +{\frac {n\alpha }{2}}\right)}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}}
sec
x
±
tan
x
=
tan
(
π
4
±
x
2
)
.
{\displaystyle \sec x\pm \tan x=\tan \left({\frac {\pi }{4}}\pm {\frac {x}{2}}\right).}
Indentitas trigonometrik Lagrange[6] [7]
∑
n
=
1
N
sin
(
n
θ
)
=
1
2
cot
θ
2
−
cos
(
(
N
+
1
2
)
θ
)
2
sin
(
θ
2
)
∑
n
=
1
N
cos
(
n
θ
)
=
−
1
2
+
sin
(
(
N
+
1
2
)
θ
)
2
sin
(
θ
2
)
{\displaystyle {\begin{aligned}\sum _{n=1}^{N}\sin(n\theta )&={\frac {1}{2}}\cot {\frac {\theta }{2}}-{\frac {\cos \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\\[5pt]\sum _{n=1}^{N}\cos(n\theta )&=-{\frac {1}{2}}+{\frac {\sin \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\end{aligned}}}
Fungsi pada nilai x , adalah Dirichlet kernel .
1
+
2
cos
x
+
2
cos
(
2
x
)
+
2
cos
(
3
x
)
+
⋯
+
2
cos
(
n
x
)
=
sin
(
(
n
+
1
2
)
x
)
sin
(
x
2
)
.
{\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {x}{2}}\right)}}.}
Lebih dari dua sinusoids
Generalisasi nya adalah[8]
∑
i
a
i
sin
(
x
+
θ
i
)
=
a
sin
(
x
+
θ
)
,
{\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}
Darimana
a
2
=
∑
i
,
j
a
i
a
j
cos
(
θ
i
−
θ
j
)
{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}
Dan
tan
θ
=
∑
i
a
i
sin
θ
i
∑
i
a
i
cos
θ
i
.
{\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.}
Perbedaan Produk dan Deret
Produk ke deret[9]
2
cos
θ
cos
φ
=
cos
(
θ
−
φ
)
+
cos
(
θ
+
φ
)
{\displaystyle 2\cos \theta \cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi )}}
2
sin
θ
sin
φ
=
cos
(
θ
−
φ
)
−
cos
(
θ
+
φ
)
{\displaystyle 2\sin \theta \sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi )}}
2
sin
θ
cos
φ
=
sin
(
θ
+
φ
)
+
sin
(
θ
−
φ
)
{\displaystyle 2\sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi )}}
2
cos
θ
sin
φ
=
sin
(
θ
+
φ
)
−
sin
(
θ
−
φ
)
{\displaystyle 2\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi )}}
tan
θ
tan
φ
=
cos
(
θ
−
φ
)
−
cos
(
θ
+
φ
)
cos
(
θ
−
φ
)
+
cos
(
θ
+
φ
)
{\displaystyle \tan \theta \tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}
∏
k
=
1
n
cos
θ
k
=
1
2
n
∑
e
∈
S
cos
(
e
1
θ
1
+
⋯
+
e
n
θ
n
)
where
S
=
{
1
,
−
1
}
n
{\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{where }}S=\{1,-1\}^{n}\end{aligned}}}
Deret ke Produk[10]
sin
θ
±
sin
φ
=
2
sin
(
θ
±
φ
2
)
cos
(
θ
∓
φ
2
)
{\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}
cos
θ
+
cos
φ
=
2
cos
(
θ
+
φ
2
)
cos
(
θ
−
φ
2
)
{\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}
cos
θ
−
cos
φ
=
−
2
sin
(
θ
+
φ
2
)
sin
(
θ
−
φ
2
)
{\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2}}\right)\sin \left({\frac {\theta -\varphi }{2}}\right)}
Fungsi produk terhingga pada trigonometri
Dalam bentuk integrasi dari nilai n , m adalah
∏
k
=
1
n
(
2
a
+
2
cos
(
2
π
k
m
n
+
x
)
)
=
2
(
T
n
(
a
)
+
(
−
1
)
n
+
m
cos
(
n
x
)
)
{\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)}
Darimana Tn adalah Chebyshev polinomial .
Dalam bentuk fungsi sinus
∏
k
=
1
n
−
1
sin
(
k
π
n
)
=
n
2
n
−
1
.
{\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.}
Dari hasil nilai generalisasi [11]
sin
(
n
x
)
=
2
n
−
1
∏
k
=
0
n
−
1
sin
(
x
+
k
π
n
)
.
{\displaystyle \sin(nx)=2^{n-1}\prod _{k=0}^{n-1}\sin \left(x+{\frac {k\pi }{n}}\right).}
Teorema pleotemy
sin
(
w
+
x
)
sin
(
x
+
y
)
=
sin
(
x
+
y
)
sin
(
y
+
z
)
(trivial)
=
sin
(
y
+
z
)
sin
(
z
+
w
)
(trivial)
=
sin
(
z
+
w
)
sin
(
w
+
x
)
(trivial)
=
sin
w
sin
y
+
sin
x
sin
z
.
(significant)
{\displaystyle {\begin{aligned}\sin(w+x)\sin(x+y)&=\sin(x+y)\sin(y+z)&{\text{(trivial)}}\\&=\sin(y+z)\sin(z+w)&{\text{(trivial)}}\\&=\sin(z+w)\sin(w+x)&{\text{(trivial)}}\\&=\sin w\sin y+\sin x\sin z.&{\text{(significant)}}\end{aligned}}}
Referensi
^ Abramowitz and Stegun, p. 80, 4.4.26–31
^ Abramowitz and Stegun, p. 75, 4.3.89–90
^ Abramowitz and Stegun, p. 85, 4.5.68–69
^ Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
^ Knapp, Michael P. "Sines and Cosines of Angles in Arithmetic Progression" (PDF) . ^
Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". American Journal of Physics . 21 (2): 140. Bibcode :1953AmJPh..21..140M . doi :10.1119/1.1933371 .
^
Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". Handbook of Mathematical Formulas and Integrals (edisi ke-4th). Academic Press. ISBN 978-0-12-374288-9 .
^ Kesalahan pengutipan: Tag <ref> tidak sah;
tidak ditemukan teks untuk ref bernama ReferenceB ^ Abramowitz and Stegun, p. 72, 4.3.31–33
^ Abramowitz and Stegun, p. 72, 4.3.34–39
^ "Product Identity Multiple Angle" .
![{\displaystyle {\begin{aligned}&\sin \varphi +\sin(\varphi +\alpha )+\sin(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\sin(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \sin \left(\varphi +{\frac {n\alpha }{2}}\right)}{\sin {\frac {\alpha }{2}}}}\quad {\text{dan}}\\[10pt]&\cos \varphi +\cos(\varphi +\alpha )+\cos(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\cos(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \cos \left(\varphi +{\frac {n\alpha }{2}}\right)}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aca78751d8093f02bd0c4f956d51f0294664553)
![{\displaystyle {\begin{aligned}\sum _{n=1}^{N}\sin(n\theta )&={\frac {1}{2}}\cot {\frac {\theta }{2}}-{\frac {\cos \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\\[5pt]\sum _{n=1}^{N}\cos(n\theta )&=-{\frac {1}{2}}+{\frac {\sin \left(\left(N+{\frac {1}{2}}\right)\theta \right)}{2\sin \left({\frac {\theta }{2}}\right)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db6f6c985a9f5be2208b7618a99495233321fa02)
![{\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{where }}S=\{1,-1\}^{n}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a86e836e8111fd80b7c225d1c4394bf4ac8459e5)
