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Things I need to memorize:
Many identities exist which interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is always 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem . In symbolic form, the Pythagorean identity reads,
(
sin
x
)
2
+
(
cos
x
)
2
=
1
,
{\displaystyle \left(\sin x\right)^{2}+\left(\cos x\right)^{2}=1\,,}
which is more commonly written with the exponent "two" next to the sine and cosine symbol:
sin
2
x
+
cos
2
x
=
1
.
{\displaystyle \sin ^{2}x+\cos ^{2}x=1\,.}
Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments which go back to Ptolemy ; one can also produce them algebraically using Euler's formula.
sin
(
x
+
y
)
=
sin
x
cos
y
+
cos
x
sin
y
,
{\displaystyle \sin \left(x+y\right)=\sin x\cos y+\cos x\sin y\,,}
cos
(
x
+
y
)
=
cos
x
cos
y
−
sin
x
sin
y
,
{\displaystyle \cos \left(x+y\right)=\cos x\cos y-\sin x\sin y\,,}
sin
(
x
−
y
)
=
sin
x
cos
y
−
cos
x
sin
y
,
{\displaystyle \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y\,,}
cos
(
x
−
y
)
=
cos
x
cos
y
+
sin
x
sin
y
.
{\displaystyle \cos \left(x-y\right)=\cos x\cos y+\sin x\sin y\,.}
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulas.
Each trigonometric function in terms of the other five. [ 1]
Function
sin
cos
tan
csc
sec
cot
sin
θ
=
{\displaystyle \sin \theta =}
sin
θ
{\displaystyle \sin \theta \ }
1
−
cos
2
θ
{\displaystyle {\sqrt {1-\cos ^{2}\theta }}}
tan
θ
1
+
tan
2
θ
{\displaystyle {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}
1
csc
θ
{\displaystyle {\frac {1}{\csc \theta }}}
sec
2
θ
−
1
sec
θ
{\displaystyle {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}
1
1
+
cot
2
θ
{\displaystyle {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}
cos
θ
=
{\displaystyle \cos \theta =}
1
−
sin
2
θ
{\displaystyle {\sqrt {1-\sin ^{2}\theta }}}
cos
θ
{\displaystyle \cos \theta \ }
1
1
+
tan
2
θ
{\displaystyle {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}
csc
2
θ
−
1
csc
θ
{\displaystyle {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}
1
sec
θ
{\displaystyle {\frac {1}{\sec \theta }}}
cot
θ
1
+
cot
2
θ
{\displaystyle {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}
tan
θ
=
{\displaystyle \tan \theta =}
sin
θ
1
−
sin
2
θ
{\displaystyle {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}
1
−
cos
2
θ
cos
θ
{\displaystyle {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}
tan
θ
{\displaystyle \tan \theta \ }
1
csc
2
θ
−
1
{\displaystyle {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}
sec
2
θ
−
1
{\displaystyle {\sqrt {\sec ^{2}\theta -1}}}
1
cot
θ
{\displaystyle {\frac {1}{\cot \theta }}}
csc
θ
=
{\displaystyle \csc \theta =}
1
sin
θ
{\displaystyle {1 \over \sin \theta }}
1
1
−
cos
2
θ
{\displaystyle {1 \over {\sqrt {1-\cos ^{2}\theta }}}}
1
+
tan
2
θ
tan
θ
{\displaystyle {{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }}
csc
θ
{\displaystyle \csc \theta \ }
sec
θ
sec
2
θ
−
1
{\displaystyle {\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}}
1
+
cot
2
θ
{\displaystyle {\sqrt {1+\cot ^{2}\theta }}}
sec
θ
=
{\displaystyle \sec \theta =}
1
1
−
sin
2
θ
{\displaystyle {1 \over {\sqrt {1-\sin ^{2}\theta }}}}
1
cos
θ
{\displaystyle {1 \over \cos \theta }}
1
+
tan
2
θ
{\displaystyle {\sqrt {1+\tan ^{2}\theta }}}
csc
θ
csc
2
θ
−
1
{\displaystyle {\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}}
sec
θ
{\displaystyle \sec \theta \ }
1
+
cot
2
θ
cot
θ
{\displaystyle {{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }}
cot
θ
=
{\displaystyle \cot \theta =}
1
−
sin
2
θ
sin
θ
{\displaystyle {{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }}
cos
θ
1
−
cos
2
θ
{\displaystyle {\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}}
1
tan
θ
{\displaystyle {1 \over \tan \theta }}
csc
2
θ
−
1
{\displaystyle {\sqrt {\csc ^{2}\theta -1}}}
1
sec
2
θ
−
1
{\displaystyle {1 \over {\sqrt {\sec ^{2}\theta -1}}}}
cot
θ
{\displaystyle \cot \theta \ }
For integrals and derivatives of trigonometric functions, see the relevant sections of table of derivatives , table of integrals , and list of integrals of trigonometric functions . Below is the list of the derivatives and integrals of the six basic trigonometric functions.
f
(
x
)
{\displaystyle \ \ \ \ f(x)}
d
d
x
f
(
x
)
{\displaystyle {\frac {d}{dx}}f(x)}
∫
f
(
x
)
d
x
{\displaystyle \int f(x)\,dx}
sin
x
{\displaystyle \,\ \sin x}
cos
x
{\displaystyle \,\ \cos x}
−
cos
x
+
C
{\displaystyle \,\ -\cos x+C}
cos
x
{\displaystyle \,\ \cos x}
−
sin
x
{\displaystyle \,\ -\sin x}
sin
x
+
C
{\displaystyle \,\ \sin x+C}
tan
x
{\displaystyle \,\ \tan x}
sec
2
x
{\displaystyle \,\ \sec ^{2}x}
−
ln
|
cos
x
|
+
C
{\displaystyle -\ln \left|\cos x\right|+C}
cot
x
{\displaystyle \,\ \cot x}
−
csc
2
x
{\displaystyle \,\ -\csc ^{2}x}
ln
|
sin
x
|
+
C
{\displaystyle \ln \left|\sin x\right|+C}
sec
x
{\displaystyle \,\ \sec x}
sec
x
tan
x
{\displaystyle \,\ \sec {x}\tan {x}}
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \ln \left|\sec x+\tan x\right|+C}
csc
x
{\displaystyle \,\ \csc x}
−
csc
x
cot
x
{\displaystyle \,\ -\csc {x}\cot {x}}
−
ln
|
csc
x
+
cot
x
|
+
C
{\displaystyle -\ln \left|\csc x+\cot x\right|+C}
These can be shown by using either the sum and difference identities or the multiple-angle formulae.
Double-angle formulae[ 2] [ 3]
sin
2
θ
=
2
sin
θ
cos
θ
=
2
tan
θ
1
+
tan
2
θ
{\displaystyle {\begin{aligned}\sin 2\theta &=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}
cos
2
θ
=
cos
2
θ
−
sin
2
θ
=
2
cos
2
θ
−
1
=
1
−
2
sin
2
θ
=
1
−
tan
2
θ
1
+
tan
2
θ
{\displaystyle {\begin{aligned}\cos 2\theta &=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}
tan
2
θ
=
2
tan
θ
1
−
tan
2
θ
{\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}\,}
cot
2
θ
=
cot
2
θ
−
1
2
cot
θ
{\displaystyle \cot 2\theta ={\frac {\cot ^{2}\theta -1}{2\cot \theta }}\,}
Triple-angle formulae[ 4] [ 5]
sin
3
θ
=
3
sin
θ
−
4
sin
3
θ
{\displaystyle \sin 3\theta =3\sin \theta -4\sin ^{3}\theta \,}
cos
3
θ
=
4
cos
3
θ
−
3
cos
θ
{\displaystyle \cos 3\theta =4\cos ^{3}\theta -3\cos \theta \,}
tan
3
θ
=
3
tan
θ
−
tan
3
θ
1
−
3
tan
2
θ
{\displaystyle \tan 3\theta ={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}
cot
3
θ
=
3
cot
θ
−
cot
3
θ
1
−
3
cot
2
θ
{\displaystyle \cot 3\theta ={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}
Half-angle formulae[ 6] [ 7]
sin
θ
2
=
±
1
−
cos
θ
2
{\displaystyle \sin {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}}
cos
θ
2
=
±
1
+
cos
θ
2
{\displaystyle \cos {\tfrac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}}}
tan
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}\tan {\tfrac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {1-\cos \theta \over 1+\cos \theta }}\\&={\frac {\sin \theta }{1+\cos \theta }}\\&={\frac {1-\cos \theta }{\sin \theta }}\end{aligned}}}
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}\cot {\tfrac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {1+\cos \theta \over 1-\cos \theta }}\\&={\frac {\sin \theta }{1-\cos \theta }}\\&={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}
See also Tangent half-angle formula .
^ Abramowitz and Stegun, p. 73, 4.3.45
^ Abramowitz and Stegun, p. 72, 4.3.24–26
^ Weisstein, Eric W. "Double-Angle Formulas" . MathWorld ^ Abramowitz and Stegun, p. 72, 4.3.27–28
^ Weisstein, Eric W. "Multiple-Angle Formulas" . MathWorld ^ Abramowitz and Stegun, p. 72, 4.3.20–22
^ Weisstein, Eric W. "Half-Angle Formulas" . MathWorld
