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Engineering Tables
sin
2
θ
+
cos
2
θ
=
1
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}
1
+
tan
2
θ
=
sec
2
θ
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
sin
(
π
2
−
θ
)
=
cos
θ
{\displaystyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta }
cos
(
π
2
−
θ
)
=
sin
θ
{\displaystyle \cos \left({\frac {\pi }{2}}-\theta \right)=\sin \theta }
sec
(
π
2
−
θ
)
=
csc
θ
{\displaystyle \sec \left({\frac {\pi }{2}}-\theta \right)=\csc \theta }
csc
(
π
2
−
θ
)
=
sec
θ
{\displaystyle \csc \left({\frac {\pi }{2}}-\theta \right)=\sec \theta }
sin
(
−
θ
)
=
−
sin
θ
{\displaystyle \sin \left(-\theta \right)=-\sin \theta }
cos
(
−
θ
)
=
cos
θ
{\displaystyle \cos \left(-\theta \right)=\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 \left({\frac {\alpha +\beta }{2}}\right)\cos \left({\frac {\alpha -\beta }{2}}\right)}
sin
α
−
sin
β
=
2
cos
(
α
+
β
2
)
sin
(
α
−
β
2
)
{\displaystyle \sin \alpha -\sin \beta =2\cos \left({\frac {\alpha +\beta }{2}}\right)\sin \left({\frac {\alpha -\beta }{2}}\right)}
cos
α
+
cos
β
=
2
cos
(
α
+
β
2
)
cos
(
α
−
β
2
)
{\displaystyle \cos \alpha +\cos \beta =2\cos \left({\frac {\alpha +\beta }{2}}\right)\cos \left({\frac {\alpha -\beta }{2}}\right)}
cos
α
−
cos
β
=
−
2
sin
(
α
+
β
2
)
sin
(
α
−
β
2
)
{\displaystyle \cos \alpha -\cos \beta =-2\sin \left({\frac {\alpha +\beta }{2}}\right)\sin \left({\frac {\alpha -\beta }{2}}\right)}
sin
α
sin
β
=
1
2
[
cos
(
α
−
β
)
−
cos
(
α
+
β
)
]
{\displaystyle \sin \alpha \sin \beta ={\frac {1}{2}}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right]}
cos
α
cos
β
=
1
2
[
cos
(
α
−
β
)
+
cos
(
α
+
β
)
]
{\displaystyle \cos \alpha \cos \beta ={\frac {1}{2}}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right]}
sin
α
cos
β
=
1
2
[
sin
(
α
+
β
)
+
sin
(
α
−
β
)
]
{\displaystyle \sin \alpha \cos \beta ={\frac {1}{2}}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right]}
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}}}
e
−
j
θ
=
cos
θ
−
j
sin
θ
{\displaystyle e^{-j\theta }=\cos \theta -j\sin \theta }
sin
θ
=
e
j
θ
−
e
−
j
θ
2
j
{\displaystyle \sin \theta ={\frac {e^{j\theta }-e^{-j\theta }}{2j}}}
tan
(
π
2
−
θ
)
=
cot
θ
{\displaystyle \tan \left({\frac {\pi }{2}}-\theta \right)=\cot \theta }
cot
(
π
2
−
θ
)
=
tan
θ
{\displaystyle \cot \left({\frac {\pi }{2}}-\theta \right)=\tan \theta }
tan
(
−
θ
)
=
−
tan
θ
{\displaystyle \tan(-\theta )=-\tan \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 }}}
tan
2
θ
=
2
tan
θ
1
−
tan
2
θ
{\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}