Engineering Handbook/Mathematics/Trigonometry
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Engineering Handbook
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Mathematics
Trigonometry Identities Table
sin
2
+
cos
2
=
1
{\displaystyle \sin ^{2}+\cos ^{2}=1}
1
+
tan
2
=
sec
2
{\displaystyle 1+\tan ^{2}=\sec ^{2}}
sin
(
π
2
−
θ
)
=
cos
θ
{\displaystyle \sin({\frac {\pi }{2}}-\theta )=\cos \theta }
cos
(
π
2
−
θ
)
=
sin
θ
{\displaystyle \cos({\frac {\pi }{2}}-\theta )=\sin \theta }
sec
(
π
2
−
θ
)
=
csc
θ
{\displaystyle \sec({\frac {\pi }{2}}-\theta )=\csc \theta }
csc
(
π
2
−
θ
)
=
sec
θ
{\displaystyle \csc({\frac {\pi }{2}}-\theta )=\sec \theta }
sin
(
−
θ
)
=
−
sin
θ
{\displaystyle \sin(-\theta )=-\sin \theta }
cos
(
−
θ
)
=
cos
θ
{\displaystyle \cos(-\theta )=\cos \theta }
sin
2
θ
=
2
sin
θ
cos
θ
{\displaystyle \sin 2\theta =2\sin \theta \cos \theta }
cos
2
θ
=
cos
2
−
sin
2
=
2
cos
2
θ
−
1
=
1
−
2
sin
2
θ
{\displaystyle \cos 2\theta =\cos ^{2}-\sin ^{2}=2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }
sin
2
θ
=
1
−
cos
2
θ
2
{\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}}
cos
2
θ
=
1
+
cos
2
θ
2
{\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}}
sin
α
+
sin
β
=
2
sin
(
α
+
β
2
)
cos
(
α
−
β
2
)
{\displaystyle \sin \alpha +\sin \beta =2\sin({\frac {\alpha +\beta }{2}})\cos({\frac {\alpha -\beta }{2}})}
sin
α
−
sin
β
=
2
cos
(
α
+
β
2
)
sin
(
α
−
β
2
)
{\displaystyle \sin \alpha -\sin \beta =2\cos({\frac {\alpha +\beta }{2}})\sin({\frac {\alpha -\beta }{2}})}
cos
α
+
cos
β
=
2
cos
(
α
+
β
2
)
cos
(
α
−
β
2
)
{\displaystyle \cos \alpha +\cos \beta =2\cos({\frac {\alpha +\beta }{2}})\cos({\frac {\alpha -\beta }{2}})}
cos
α
−
cos
β
=
−
2
sin
(
α
+
β
2
)
sin
(
α
−
β
2
)
{\displaystyle \cos \alpha -\cos \beta =-2\sin({\frac {\alpha +\beta }{2}})\sin({\frac {\alpha -\beta }{2}})}
sin
α
sin
β
=
1
2
[
cos
(
α
−
β
)
−
cos
(
α
+
β
)
]
{\displaystyle \sin \alpha \sin \beta ={\frac {1}{2}}[\cos(\alpha -\beta )-\cos(\alpha +\beta )]}
cos
α
cos
β
=
1
2
[
cos
(
α
−
β
)
+
cos
(
α
+
β
)
]
{\displaystyle \cos \alpha \cos \beta ={\frac {1}{2}}[\cos(\alpha -\beta )+\cos(\alpha +\beta )]}
sin
α
cos
β
=
1
2
[
sin
(
α
+
β
)
+
sin
(
α
−
β
)
]
{\displaystyle \sin \alpha \cos \beta ={\frac {1}{2}}[\sin(\alpha +\beta )+\sin(\alpha -\beta )]}
1
+
cot
2
=
csc
2
{\displaystyle 1+\cot ^{2}=\csc ^{2}}
e
j
θ
=
cos
θ
+
j
sin
θ
{\displaystyle e^{j\theta }=\cos \theta +j\sin \theta }
cos
θ
=
e
j
θ
+
e
−
j
θ
2
{\displaystyle \cos \theta ={\frac {e^{j\theta }+e^{-j\theta }}{2}}}
tan
(
π
2
−
θ
)
=
cot
θ
{\displaystyle \tan({\frac {\pi }{2}}-\theta )=\cot \theta }
cot
(
π
2
−
θ
)
=
tan
θ
{\displaystyle \cot({\frac {\pi }{2}}-\theta )=\tan \theta }
tan
(
−
θ
)
=
−
tan
θ
{\displaystyle \tan(-\theta )=-\tan \theta }
tan
2
θ
=
2
tan
θ
1
−
t
a
n
2
θ
{\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-tan^{2}\theta }}}
tan
2
θ
=
1
−
cos
2
θ
1
+
cos
2
θ
{\displaystyle \tan ^{2}\theta ={\frac {1-\cos 2\theta }{1+\cos 2\theta }}}
sin
θ
=
e
j
θ
−
e
−
j
θ
j
2
{\displaystyle \sin \theta ={\frac {e^{j\theta }-e^{-j\theta }}{j2}}}
References
Trigonometric Identities
![{\displaystyle \sin \alpha \sin \beta ={\frac {1}{2}}[\cos(\alpha -\beta )-\cos(\alpha +\beta )]}](../../_assets_/eb734a37dd21ce173a46342d1cc64c92/910a60c63ebad0da7066fe1db46b6f12fff6c8b2.svg)
![{\displaystyle \cos \alpha \cos \beta ={\frac {1}{2}}[\cos(\alpha -\beta )+\cos(\alpha +\beta )]}](../../_assets_/eb734a37dd21ce173a46342d1cc64c92/9a72aa9db653d183950a4e7b565af533ea6c169b.svg)
![{\displaystyle \sin \alpha \cos \beta ={\frac {1}{2}}[\sin(\alpha +\beta )+\sin(\alpha -\beta )]}](../../_assets_/eb734a37dd21ce173a46342d1cc64c92/0b614639caba82f3c3931406612d821cd5234d35.svg)
