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List of trigonometric identities

Updated: Wikipedia source

List of trigonometric identities

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Tables

Each trigonometric function in terms of each of the other five. · Pythagorean identities
sin ⁡ θ = {\displaystyle \sin \theta =}
sin ⁡ θ = {\displaystyle \sin \theta =}
in terms of → ↓
sin ⁡ θ = {\displaystyle \sin \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
sin ⁡ θ {\displaystyle \sin \theta }
csc ⁡ θ {\displaystyle \csc \theta }
1 csc ⁡ θ {\displaystyle {\frac {1}{\csc \theta }}}
cos ⁡ θ {\displaystyle \cos \theta }
± 1 − cos 2 ⁡ θ {\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}
sec ⁡ θ {\displaystyle \sec \theta }
± sec 2 ⁡ θ − 1 sec ⁡ θ {\displaystyle \pm {\f
tan ⁡ θ {\displaystyle \tan \theta }
± tan ⁡ θ 1 + tan 2 ⁡ θ {\displaystyle \pm {\f
cot ⁡ θ {\displaystyle \cot \theta }
± 1 1 + cot 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}
csc ⁡ θ = {\displaystyle \csc \theta =}
csc ⁡ θ = {\displaystyle \csc \theta =}
in terms of → ↓
csc ⁡ θ = {\displaystyle \csc \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
1 sin ⁡ θ {\displaystyle {\frac {1}{\sin \theta }}}
csc ⁡ θ {\displaystyle \csc \theta }
csc ⁡ θ {\displaystyle \csc \theta }
cos ⁡ θ {\displaystyle \cos \theta }
± 1 1 − cos 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}
sec ⁡ θ {\displaystyle \sec \theta }
± sec ⁡ θ sec 2 ⁡ θ − 1 {\displaystyle \pm {\f
tan ⁡ θ {\displaystyle \tan \theta }
± 1 + tan 2 ⁡ θ tan ⁡ θ {\displaystyle \pm {\f
cot ⁡ θ {\displaystyle \cot \theta }
± 1 + cot 2 ⁡ θ {\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}
cos ⁡ θ = {\displaystyle \cos \theta =}
cos ⁡ θ = {\displaystyle \cos \theta =}
in terms of → ↓
cos ⁡ θ = {\displaystyle \cos \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
± 1 − sin 2 ⁡ θ {\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}
csc ⁡ θ {\displaystyle \csc \theta }
± csc 2 ⁡ θ − 1 csc ⁡ θ {\displaystyle \pm {\f
cos ⁡ θ {\displaystyle \cos \theta }
cos ⁡ θ {\displaystyle \cos \theta }
sec ⁡ θ {\displaystyle \sec \theta }
1 sec ⁡ θ {\displaystyle {\frac {1}{\sec \theta }}}
tan ⁡ θ {\displaystyle \tan \theta }
± 1 1 + tan 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}
cot ⁡ θ {\displaystyle \cot \theta }
± cot ⁡ θ 1 + cot 2 ⁡ θ {\displaystyle \pm {\f
sec ⁡ θ = {\displaystyle \sec \theta =}
sec ⁡ θ = {\displaystyle \sec \theta =}
in terms of → ↓
sec ⁡ θ = {\displaystyle \sec \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
± 1 1 − sin 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}
csc ⁡ θ {\displaystyle \csc \theta }
± csc ⁡ θ csc 2 ⁡ θ − 1 {\displaystyle \pm {\f
cos ⁡ θ {\displaystyle \cos \theta }
1 cos ⁡ θ {\displaystyle {\frac {1}{\cos \theta }}}
sec ⁡ θ {\displaystyle \sec \theta }
sec ⁡ θ {\displaystyle \sec \theta }
tan ⁡ θ {\displaystyle \tan \theta }
± 1 + tan 2 ⁡ θ {\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}
cot ⁡ θ {\displaystyle \cot \theta }
± 1 + cot 2 ⁡ θ cot ⁡ θ {\displaystyle \pm {\f
tan ⁡ θ = {\displaystyle \tan \theta =}
tan ⁡ θ = {\displaystyle \tan \theta =}
in terms of → ↓
tan ⁡ θ = {\displaystyle \tan \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
± sin ⁡ θ 1 − sin 2 ⁡ θ {\displaystyle \pm {\f
csc ⁡ θ {\displaystyle \csc \theta }
± 1 csc 2 ⁡ θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}
cos ⁡ θ {\displaystyle \cos \theta }
± 1 − cos 2 ⁡ θ cos ⁡ θ {\displaystyle \pm {\f
sec ⁡ θ {\displaystyle \sec \theta }
± sec 2 ⁡ θ − 1 {\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}
tan ⁡ θ {\displaystyle \tan \theta }
tan ⁡ θ {\displaystyle \tan \theta }
cot ⁡ θ {\displaystyle \cot \theta }
1 cot ⁡ θ {\displaystyle {\frac {1}{\cot \theta }}}
cot ⁡ θ = {\displaystyle \cot \theta =}
cot ⁡ θ = {\displaystyle \cot \theta =}
in terms of → ↓
cot ⁡ θ = {\displaystyle \cot \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
± 1 − sin 2 ⁡ θ sin ⁡ θ {\displaystyle \pm {\f
csc ⁡ θ {\displaystyle \csc \theta }
± csc 2 ⁡ θ − 1 {\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}
cos ⁡ θ {\displaystyle \cos \theta }
± cos ⁡ θ 1 − cos 2 ⁡ θ {\displaystyle \pm {\f
sec ⁡ θ {\displaystyle \sec \theta }
± 1 sec 2 ⁡ θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}
tan ⁡ θ {\displaystyle \tan \theta }
1 tan ⁡ θ {\displaystyle {\frac {1}{\tan \theta }}}
cot ⁡ θ {\displaystyle \cot \theta }
cot ⁡ θ {\displaystyle \cot \theta }
in terms of → ↓
sin ⁡ θ {\displaystyle \sin \theta }
csc ⁡ θ {\displaystyle \csc \theta }
cos ⁡ θ {\displaystyle \cos \theta }
sec ⁡ θ {\displaystyle \sec \theta }
tan ⁡ θ {\displaystyle \tan \theta }
cot ⁡ θ {\displaystyle \cot \theta }
sin ⁡ θ = {\displaystyle \sin \theta =}
sin ⁡ θ {\displaystyle \sin \theta }
1 csc ⁡ θ {\displaystyle {\frac {1}{\csc \theta }}}
± 1 − cos 2 ⁡ θ {\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}
± sec 2 ⁡ θ − 1 sec ⁡ θ {\displaystyle \pm {\f
± tan ⁡ θ 1 + tan 2 ⁡ θ {\displaystyle \pm {\f
± 1 1 + cot 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}
csc ⁡ θ = {\displaystyle \csc \theta =}
1 sin ⁡ θ {\displaystyle {\frac {1}{\sin \theta }}}
csc ⁡ θ {\displaystyle \csc \theta }
± 1 1 − cos 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}
± sec ⁡ θ sec 2 ⁡ θ − 1 {\displaystyle \pm {\f
± 1 + tan 2 ⁡ θ tan ⁡ θ {\displaystyle \pm {\f
± 1 + cot 2 ⁡ θ {\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}
cos ⁡ θ = {\displaystyle \cos \theta =}
± 1 − sin 2 ⁡ θ {\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}
± csc 2 ⁡ θ − 1 csc ⁡ θ {\displaystyle \pm {\f
cos ⁡ θ {\displaystyle \cos \theta }
1 sec ⁡ θ {\displaystyle {\frac {1}{\sec \theta }}}
± 1 1 + tan 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}
± cot ⁡ θ 1 + cot 2 ⁡ θ {\displaystyle \pm {\f
sec ⁡ θ = {\displaystyle \sec \theta =}
± 1 1 − sin 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}
± csc ⁡ θ csc 2 ⁡ θ − 1 {\displaystyle \pm {\f
1 cos ⁡ θ {\displaystyle {\frac {1}{\cos \theta }}}
sec ⁡ θ {\displaystyle \sec \theta }
± 1 + tan 2 ⁡ θ {\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}
± 1 + cot 2 ⁡ θ cot ⁡ θ {\displaystyle \pm {\f
tan ⁡ θ = {\displaystyle \tan \theta =}
± sin ⁡ θ 1 − sin 2 ⁡ θ {\displaystyle \pm {\f
± 1 csc 2 ⁡ θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}
± 1 − cos 2 ⁡ θ cos ⁡ θ {\displaystyle \pm {\f
± sec 2 ⁡ θ − 1 {\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}
tan ⁡ θ {\displaystyle \tan \theta }
1 cot ⁡ θ {\displaystyle {\frac {1}{\cot \theta }}}
cot ⁡ θ = {\displaystyle \cot \theta =}
± 1 − sin 2 ⁡ θ sin ⁡ θ {\displaystyle \pm {\f
± csc 2 ⁡ θ − 1 {\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}
± cos ⁡ θ 1 − cos 2 ⁡ θ {\displaystyle \pm {\f
± 1 sec 2 ⁡ θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}
1 tan ⁡ θ {\displaystyle {\frac {1}{\tan \theta }}}
cot ⁡ θ {\displaystyle \cot \theta }
sin ⁡ ( − θ ) = − sin ⁡ θ {\displaystyle \sin(-\theta )=-\sin \theta }
sin ⁡ ( − θ ) = − sin ⁡ θ {\displaystyle \sin(-\theta )=-\sin \theta }
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
sin ⁡ ( − θ ) = − sin ⁡ θ {\displaystyle \sin(-\theta )=-\sin \theta }
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
sin ⁡ ( π 2 − θ ) = cos ⁡ θ {\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \rig
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
sin ⁡ ( π − θ ) = + sin ⁡ θ {\displaystyle \sin(\pi -\theta )=+\sin \theta }
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
sin ⁡ ( 3 π 2 − θ ) = − cos ⁡ θ
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
sin ⁡ ( 2 π − θ ) = − sin ⁡ ( θ ) = sin ⁡ ( − θ ) {\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}
cos ⁡ ( − θ ) = + cos ⁡ θ {\displaystyle \cos(-\theta )=+\cos \theta }
cos ⁡ ( − θ ) = + cos ⁡ θ {\displaystyle \cos(-\theta )=+\cos \theta }
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
cos ⁡ ( − θ ) = + cos ⁡ θ {\displaystyle \cos(-\theta )=+\cos \theta }
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
cos ⁡ ( π 2 − θ ) = sin ⁡ θ {\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \rig
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
cos ⁡ ( π − θ ) = − cos ⁡ θ {\displaystyle \cos(\pi -\theta )=-\cos \theta }
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
cos ⁡ ( 3 π 2 − θ ) = − sin ⁡ θ
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
cos ⁡ ( 2 π − θ ) = + cos ⁡ ( θ ) = cos ⁡ ( − θ ) {\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}
tan ⁡ ( − θ ) = − tan ⁡ θ {\displaystyle \tan(-\theta )=-\tan \theta }
tan ⁡ ( − θ ) = − tan ⁡ θ {\displaystyle \tan(-\theta )=-\tan \theta }
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
tan ⁡ ( − θ ) = − tan ⁡ θ {\displaystyle \tan(-\theta )=-\tan \theta }
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
tan ⁡ ( π 2 − θ ) = cot ⁡ θ {\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \rig
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
tan ⁡ ( π − θ ) = − tan ⁡ θ {\displaystyle \tan(\pi -\theta )=-\tan \theta }
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
tan ⁡ ( 3 π 2 − θ ) = + cot ⁡ θ
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
tan ⁡ ( 2 π − θ ) = − tan ⁡ ( θ ) = tan ⁡ ( − θ ) {\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}
csc ⁡ ( − θ ) = − csc ⁡ θ {\displaystyle \csc(-\theta )=-\csc \theta }
csc ⁡ ( − θ ) = − csc ⁡ θ {\displaystyle \csc(-\theta )=-\csc \theta }
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
csc ⁡ ( − θ ) = − csc ⁡ θ {\displaystyle \csc(-\theta )=-\csc \theta }
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
csc ⁡ ( π 2 − θ ) = sec ⁡ θ {\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \rig
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
csc ⁡ ( π − θ ) = + csc ⁡ θ {\displaystyle \csc(\pi -\theta )=+\csc \theta }
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
csc ⁡ ( 3 π 2 − θ ) = − sec ⁡ θ
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
csc ⁡ ( 2 π − θ ) = − csc ⁡ ( θ ) = csc ⁡ ( − θ ) {\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}
sec ⁡ ( − θ ) = + sec ⁡ θ {\displaystyle \sec(-\theta )=+\sec \theta }
sec ⁡ ( − θ ) = + sec ⁡ θ {\displaystyle \sec(-\theta )=+\sec \theta }
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
sec ⁡ ( − θ ) = + sec ⁡ θ {\displaystyle \sec(-\theta )=+\sec \theta }
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
sec ⁡ ( π 2 − θ ) = csc ⁡ θ {\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \rig
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
sec ⁡ ( π − θ ) = − sec ⁡ θ {\displaystyle \sec(\pi -\theta )=-\sec \theta }
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
sec ⁡ ( 3 π 2 − θ ) = − csc ⁡ θ
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
sec ⁡ ( 2 π − θ ) = + sec ⁡ ( θ ) = sec ⁡ ( − θ ) {\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}
cot ⁡ ( − θ ) = − cot ⁡ θ {\displaystyle \cot(-\theta )=-\cot \theta }
cot ⁡ ( − θ ) = − cot ⁡ θ {\displaystyle \cot(-\theta )=-\cot \theta }
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
cot ⁡ ( − θ ) = − cot ⁡ θ {\displaystyle \cot(-\theta )=-\cot \theta }
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
cot ⁡ ( π 2 − θ ) = tan ⁡ θ {\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \rig
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
cot ⁡ ( π − θ ) = − cot ⁡ θ {\displaystyle \cot(\pi -\theta )=-\cot \theta }
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
cot ⁡ ( 3 π 2 − θ ) = + tan ⁡ θ
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
cot ⁡ ( 2 π − θ ) = − cot ⁡ ( θ ) = cot ⁡ ( − θ ) {\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}
θ {\displaystyle \theta } reflected in α = 0 {\displaystyle \alpha =0} odd/even identities
θ {\displaystyle \theta } reflected in α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} complementary angles
θ {\displaystyle \theta } reflected in α = π 2 {\displaystyle \alpha ={\frac {\pi }{2}}} supplementary angles
θ {\displaystyle \theta } reflected in α = 3 π 4 {\displaystyle \alpha ={\frac {3\pi }{4}}}
θ {\displaystyle \theta } reflected in α = π {\displaystyle \alpha =\pi } conjugate angles; compare to α = 0 {\displaystyle \alpha =0}
sin ⁡ ( − θ ) = − sin ⁡ θ {\displaystyle \sin(-\theta )=-\sin \theta }
sin ⁡ ( π 2 − θ ) = cos ⁡ θ {\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta ig
sin ⁡ ( π − θ ) = + sin ⁡ θ {\displaystyle \sin(\pi -\theta )=+\sin \theta }
sin ⁡ ( 3 π 2 − θ ) = − cos ⁡ θ
sin ⁡ ( 2 π − θ ) = − sin ⁡ ( θ ) = sin ⁡ ( − θ ) {\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}
cos ⁡ ( − θ ) = + cos ⁡ θ {\displaystyle \cos(-\theta )=+\cos \theta }
cos ⁡ ( π 2 − θ ) = sin ⁡ θ {\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta ig
cos ⁡ ( π − θ ) = − cos ⁡ θ {\displaystyle \cos(\pi -\theta )=-\cos \theta }
cos ⁡ ( 3 π 2 − θ ) = − sin ⁡ θ
cos ⁡ ( 2 π − θ ) = + cos ⁡ ( θ ) = cos ⁡ ( − θ ) {\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}
tan ⁡ ( − θ ) = − tan ⁡ θ {\displaystyle \tan(-\theta )=-\tan \theta }
tan ⁡ ( π 2 − θ ) = cot ⁡ θ {\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta ig
tan ⁡ ( π − θ ) = − tan ⁡ θ {\displaystyle \tan(\pi -\theta )=-\tan \theta }
tan ⁡ ( 3 π 2 − θ ) = + cot ⁡ θ
tan ⁡ ( 2 π − θ ) = − tan ⁡ ( θ ) = tan ⁡ ( − θ ) {\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}
csc ⁡ ( − θ ) = − csc ⁡ θ {\displaystyle \csc(-\theta )=-\csc \theta }
csc ⁡ ( π 2 − θ ) = sec ⁡ θ {\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta ig
csc ⁡ ( π − θ ) = + csc ⁡ θ {\displaystyle \csc(\pi -\theta )=+\csc \theta }
csc ⁡ ( 3 π 2 − θ ) = − sec ⁡ θ
csc ⁡ ( 2 π − θ ) = − csc ⁡ ( θ ) = csc ⁡ ( − θ ) {\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}
sec ⁡ ( − θ ) = + sec ⁡ θ {\displaystyle \sec(-\theta )=+\sec \theta }
sec ⁡ ( π 2 − θ ) = csc ⁡ θ {\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta ig
sec ⁡ ( π − θ ) = − sec ⁡ θ {\displaystyle \sec(\pi -\theta )=-\sec \theta }
sec ⁡ ( 3 π 2 − θ ) = − csc ⁡ θ
sec ⁡ ( 2 π − θ ) = + sec ⁡ ( θ ) = sec ⁡ ( − θ ) {\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}
cot ⁡ ( − θ ) = − cot ⁡ θ {\displaystyle \cot(-\theta )=-\cot \theta }
cot ⁡ ( π 2 − θ ) = tan ⁡ θ {\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta ig
cot ⁡ ( π − θ ) = − cot ⁡ θ {\displaystyle \cot(\pi -\theta )=-\cot \theta }
cot ⁡ ( 3 π 2 − θ ) = + tan ⁡ θ
cot ⁡ ( 2 π − θ ) = − cot ⁡ ( θ ) = cot ⁡ ( − θ ) {\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}
· Reflections, shifts, and periodicity › Shifts and periodicity
sin ⁡ ( θ ± π 2 ) = ± cos ⁡ θ {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }
sin ⁡ ( θ ± π 2 ) = ± cos ⁡ θ {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }
Shift by one quarter period orthogonal angles
sin ⁡ ( θ ± π 2 ) = ± cos ⁡ θ {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }
Shift by one half period opposite angles
sin ⁡ ( θ + π ) = − sin ⁡ θ {\displaystyle \sin(\theta +\pi )=-\sin \theta }
Shift by full periods coincident angles
sin ⁡ ( θ + k ⋅ 2 π ) = + sin ⁡ θ {\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }
Period
2 π {\displaystyle 2\pi }
cos ⁡ ( θ ± π 2 ) = ∓ sin ⁡ θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }
cos ⁡ ( θ ± π 2 ) = ∓ sin ⁡ θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }
Shift by one quarter period orthogonal angles
cos ⁡ ( θ ± π 2 ) = ∓ sin ⁡ θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }
Shift by one half period opposite angles
cos ⁡ ( θ + π ) = − cos ⁡ θ {\displaystyle \cos(\theta +\pi )=-\cos \theta }
Shift by full periods coincident angles
cos ⁡ ( θ + k ⋅ 2 π ) = + cos ⁡ θ {\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }
Period
2 π {\displaystyle 2\pi }
csc ⁡ ( θ ± π 2 ) = ± sec ⁡ θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }
csc ⁡ ( θ ± π 2 ) = ± sec ⁡ θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }
Shift by one quarter period orthogonal angles
csc ⁡ ( θ ± π 2 ) = ± sec ⁡ θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }
Shift by one half period opposite angles
csc ⁡ ( θ + π ) = − csc ⁡ θ {\displaystyle \csc(\theta +\pi )=-\csc \theta }
Shift by full periods coincident angles
csc ⁡ ( θ + k ⋅ 2 π ) = + csc ⁡ θ {\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }
Period
2 π {\displaystyle 2\pi }
sec ⁡ ( θ ± π 2 ) = ∓ csc ⁡ θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }
sec ⁡ ( θ ± π 2 ) = ∓ csc ⁡ θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }
Shift by one quarter period orthogonal angles
sec ⁡ ( θ ± π 2 ) = ∓ csc ⁡ θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }
Shift by one half period opposite angles
sec ⁡ ( θ + π ) = − sec ⁡ θ {\displaystyle \sec(\theta +\pi )=-\sec \theta }
Shift by full periods coincident angles
sec ⁡ ( θ + k ⋅ 2 π ) = + sec ⁡ θ {\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }
Period
2 π {\displaystyle 2\pi }
tan ⁡ ( θ ± π 4 ) = tan ⁡ θ ± 1 1
tan ⁡ ( θ ± π 4 ) = tan ⁡ θ ± 1 1
Shift by one quarter period orthogonal angles
tan ⁡ ( θ ± π 4 ) = tan ⁡ θ ± 1 1
Shift by one half period opposite angles
tan ⁡ ( θ + π 2 ) = − cot ⁡ θ {\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }
Shift by full periods coincident angles
tan ⁡ ( θ + k ⋅ π ) = + tan ⁡ θ {\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }
Period
π {\displaystyle \pi }
cot ⁡ ( θ ± π 4 ) = cot ⁡ θ ∓ 1 1
cot ⁡ ( θ ± π 4 ) = cot ⁡ θ ∓ 1 1
Shift by one quarter period orthogonal angles
cot ⁡ ( θ ± π 4 ) = cot ⁡ θ ∓ 1 1
Shift by one half period opposite angles
cot ⁡ ( θ + π 2 ) = − tan ⁡ θ {\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }
Shift by full periods coincident angles
cot ⁡ ( θ + k ⋅ π ) = + cot ⁡ θ {\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }
Period
π {\displaystyle \pi }
Shift by one quarter period orthogonal angles
Shift by one half period opposite angles
Shift by full periods coincident angles
Period
sin ⁡ ( θ ± π 2 ) = ± cos ⁡ θ {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }
sin ⁡ ( θ + π ) = − sin ⁡ θ {\displaystyle \sin(\theta +\pi )=-\sin \theta }
sin ⁡ ( θ + k ⋅ 2 π ) = + sin ⁡ θ {\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }
2 π {\displaystyle 2\pi }
cos ⁡ ( θ ± π 2 ) = ∓ sin ⁡ θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }
cos ⁡ ( θ + π ) = − cos ⁡ θ {\displaystyle \cos(\theta +\pi )=-\cos \theta }
cos ⁡ ( θ + k ⋅ 2 π ) = + cos ⁡ θ {\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }
2 π {\displaystyle 2\pi }
csc ⁡ ( θ ± π 2 ) = ± sec ⁡ θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }
csc ⁡ ( θ + π ) = − csc ⁡ θ {\displaystyle \csc(\theta +\pi )=-\csc \theta }
csc ⁡ ( θ + k ⋅ 2 π ) = + csc ⁡ θ {\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }
2 π {\displaystyle 2\pi }
sec ⁡ ( θ ± π 2 ) = ∓ csc ⁡ θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }
sec ⁡ ( θ + π ) = − sec ⁡ θ {\displaystyle \sec(\theta +\pi )=-\sec \theta }
sec ⁡ ( θ + k ⋅ 2 π ) = + sec ⁡ θ {\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }
2 π {\displaystyle 2\pi }
tan ⁡ ( θ ± π 4 ) = tan ⁡ θ ± 1 1
tan ⁡ ( θ + π 2 ) = − cot ⁡ θ {\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }
tan ⁡ ( θ + k ⋅ π ) = + tan ⁡ θ {\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }
π {\displaystyle \pi }
cot ⁡ ( θ ± π 4 ) = cot ⁡ θ ∓ 1 1
cot ⁡ ( θ + π 2 ) = − tan ⁡ θ {\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }
cot ⁡ ( θ + k ⋅ π ) = + cot ⁡ θ {\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }
π {\displaystyle \pi }
· Angle sum and difference identities
Cosine
Cosine
Sine
Cosine
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
cos ⁡ ( α ± β ) {\displaystyle \cos(\alpha \pm \beta )}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
cos ⁡ α cos ⁡ β ∓ sin ⁡ α sin ⁡ β {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
Tangent
Tangent
Sine
Tangent
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
tan ⁡ ( α ± β ) {\displaystyle \tan(\alpha \pm \beta )}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
tan ⁡ α ± tan ⁡ β 1 ∓ tan ⁡ α tan ⁡ β {\displaystyle {\frac {\tan \alpha \pm \t
Cosecant
Cosecant
Sine
Cosecant
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
csc ⁡ ( α ± β ) {\displaystyle \csc(\alpha \pm \beta )}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β sec ⁡ α csc ⁡ β ± csc
Secant
Secant
Sine
Secant
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
sec ⁡ ( α ± β ) {\displaystyle \sec(\alpha \pm \beta )}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β csc ⁡ α csc ⁡ β ∓ sec
Cotangent
Cotangent
Sine
Cotangent
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
cot ⁡ ( α ± β ) {\displaystyle \cot(\alpha \pm \beta )}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
cot ⁡ α cot ⁡ β ∓ 1 cot ⁡ β ± cot ⁡ α {\displaystyle {\frac {\cot \alpha \cot \
Arcsine
Arcsine
Sine
Arcsine
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arcsin ⁡ x ± arcsin ⁡ y {\displaystyle \arcsin x\pm \arcsin y}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arcsin ⁡ ( x 1 − y 2 ± y 1
Arccosine
Arccosine
Sine
Arccosine
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arccos ⁡ x ± arccos ⁡ y {\displaystyle \arccos x\pm \arccos y}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arccos ⁡ ( x y ∓ ( 1 − x 2
Arctangent
Arctangent
Sine
Arctangent
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arctan ⁡ x ± arctan ⁡ y {\displaystyle \arctan x\pm \arctan y}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arctan ⁡ ( x ± y 1 ∓ x y ) {\displaystyle \arctan \left({\frac {x\pm
Arccotangent
Arccotangent
Sine
Arccotangent
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arccot ⁡ x ± arccot ⁡ y {\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
arccot ⁡ ( x y ∓ 1 y ± x ) {\displaystyle \operatorname {arccot} \le
Sine
sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )}
= {\displaystyle =}
sin ⁡ α cos ⁡ β ± cos ⁡ α sin ⁡ β {\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }
Cosine
cos ⁡ ( α ± β ) {\displaystyle \cos(\alpha \pm \beta )}
= {\displaystyle =}
cos ⁡ α cos ⁡ β ∓ sin ⁡ α sin ⁡ β {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
Tangent
tan ⁡ ( α ± β ) {\displaystyle \tan(\alpha \pm \beta )}
= {\displaystyle =}
tan ⁡ α ± tan ⁡ β 1 ∓ tan ⁡ α tan ⁡ β {\displaystyle {\frac {\tan \alpha \pm \t
Cosecant
csc ⁡ ( α ± β ) {\displaystyle \csc(\alpha \pm \beta )}
= {\displaystyle =}
sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β sec ⁡ α csc ⁡ β ± csc
Secant
sec ⁡ ( α ± β ) {\displaystyle \sec(\alpha \pm \beta )}
= {\displaystyle =}
sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β csc ⁡ α csc ⁡ β ∓ sec
Cotangent
cot ⁡ ( α ± β ) {\displaystyle \cot(\alpha \pm \beta )}
= {\displaystyle =}
cot ⁡ α cot ⁡ β ∓ 1 cot ⁡ β ± cot ⁡ α {\displaystyle {\frac {\cot \alpha \cot \
Arcsine
arcsin ⁡ x ± arcsin ⁡
= {\displaystyle =}
arcsin ⁡ ( x 1 − y 2 ± y 1
Arccosine
arccos ⁡ x ± arccos ⁡
= {\displaystyle =}
arccos ⁡ ( x y ∓ ( 1 − x 2
Arctangent
arctan ⁡ x ± arctan ⁡
= {\displaystyle =}
arctan ⁡ ( x ± y 1 ∓ x y ) {\displaystyle \arctan \left({\frac {x\pm
Arccotangent
arccot ⁡ x ± arccot ⁡ x\pm \operatorname {arccot} y}
= {\displaystyle =}
arccot ⁡ ( x y ∓ 1 y ± x ) {\displaystyle \operatorname {arccot} \le
· Multiple-angle and half-angle formulas
de Moivre's formula, i is the imaginary unit
de Moivre's formula, i is the imaginary unit
Tn is the nth Chebyshev polynomial
de Moivre's formula, i is the imaginary unit
cos ⁡ ( n θ ) = T n ( cos ⁡ θ ) {\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}
cos ⁡ ( n θ ) + i sin ⁡ ( n θ ) = ( cos ⁡ θ + i sin ⁡ θ ) n {\displaystyle \cos(n\theta
Tn is the nth Chebyshev polynomial
cos ⁡ ( n θ ) = T n ( cos ⁡ θ ) {\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}
de Moivre's formula, i is the imaginary unit
cos ⁡ ( n θ ) + i sin ⁡ ( n θ ) = ( cos ⁡ θ + i sin ⁡ θ ) n {\displaystyle \cos(n\theta

References

  1. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
    http://www.math.ubc.ca/~cbm/aands/page_73.htm
  2. Selby 1970, p. 188
  3. Abramowitz and Stegun, p. 72, 4 –15
  4. Abramowitz and Stegun, p. 72, 4 –9
  5. Abramowitz and Stegun, p. 72, 4
  6. MathWorld
    https://mathworld.wolfram.com/TrigonometricAdditionFormulas.html
  7. Abramowitz and Stegun, p. 72, 4
  8. Abramowitz and Stegun, p. 72, 4
  9. www
    http://www.milefoot.com/math/trig/22anglesumidentities.htm
  10. Abramowitz and Stegun, p. 72, 4
  11. Abramowitz and Stegun, p. 80, 4
  12. Abramowitz and Stegun, p. 80, 4
  13. Abramowitz and Stegun, p. 80, 4
  14. Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation
    https://doi.org/10.1145%2F74540.74566
  15. Michael Hardy. (2016). "On Tangents and Secants of Infinite Sums." The American Mathematical Monthly, volume 123, number
    https://doi.org/10.4169/amer.math.monthly.123.7.701
  16. The American Mathematical Monthly
    https://doi.org/10.1080%2F00029890.2025.2459048
  17. Proceedings of the American Mathematical Society
  18. American Mathematical Monthly
    https://zenodo.org/record/1000408
  19. "Sine, Cosine, and Ptolemy's Theorem"
    https://www.cut-the-knot.org/proofs/sine_cosine.shtml
  20. MathWorld
    https://mathworld.wolfram.com/Multiple-AngleFormulas.html
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