Calculus/Table of Trigonometry

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Definitions[edit]

  • \tan(x)=\frac{\sin(x)}{\cos(x)}
  • \sec(x)=\frac{1}{\cos(x)}
  • \cot(x)=\frac{\cos(x)}{\sin(x)}=\frac{1}{\tan(x)}
  • \csc(x)=\frac{1}{\sin(x)}

Pythagorean Identities[edit]

  • \sin^2(x)+\cos^2(x)=1
  • 1+\tan^2(x)=\sec^2(x)
  • 1+\cot^2(x)=\csc^2(x)

Double Angle Identities[edit]

  • \sin(2x)=2\sin(x)\cos(x)
  • \cos(2x)=\cos^2(x)-\sin^2(x)
  • \tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}
  • \cos^2(x)=\frac{1+\cos(2x)}{2}
  • \sin^2(x)=\frac{1-\cos(2x)}{2}

Angle Sum Identities[edit]

\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)
\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)
\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)
\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)
\sin(x)+\sin(y)=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)
\sin(x)-\sin(y)=2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)
\cos(x)+\cos(y)=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)
\cos(x)-\cos(y)=-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)
\tan(x)+\tan(y)=\frac{\sin(x+y)}{\cos(x)\cos(y)}
\tan(x)-\tan(y)=\frac{\sin(x-y)}{\cos(x)\cos(y)}
\cot(x)+\cot(y)=\frac{\sin(x+y)}{\sin(x)\sin(y)}
\cot(x)-\cot(y=)\frac{-\sin(x-y)}{\sin(x)\sin(y)}

Product-to-sum identities[edit]

\cos(x)\cos(y)= \frac{\cos(x+y)+\cos(x-y)}{2}
\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x+y)}{2}
\sin(x)\cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}
\cos(x)\sin(y)=\frac{\sin(x+y)-\sin(x-y)}{2}

See also[edit]