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Trigonometric formulas

Negative angles

  1. \(\sin{(-x)}=-\sin{x}\)
  2. \(\cos{(-x)}=\cos{x}\)
  3. \(\tan{(-x)}=-\tan{x}\)
Tip

Just change \(-x\) to \(x\), and add the sign depending the sign on \(IV\) quadrant.

Compound angles

  1. \(\sin{(x+y)}=\sin{x}\cos{y}+\cos{x}\sin{y}\)

    \(\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}\)

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

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

  3. \(\tan{(x+y)}=\frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}\)

    \(\tan{(x-y)}=\frac{\tan{x}-\tan{y}}{1+\tan{x}\tan{y}}\)

  4. \(\cot{(x+y)}=\frac{\cot{x}\cot{y}-1}{\cot{x}+\cot{y}}\)

    \(\cot{(x-y)}=\frac{\cot{x}\cot{y}+1}{\cot{y}-\cot{x}}\)

Trigonometric ratios of compound angles

Check this short to understand: Trigonometric ratios of compound angles

Allied angles

Allied angles are angles whose sum or difference is a multiple of \(\frac{\pi}{2}\).

  1. \(\sin{(\frac{\pi}{2}-x)}=\cos{x}\)

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

    \(\sin{(\pi-x)}=\sin{x}\)

    \(\sin{(\pi+x)}=-\sin{x}\)

  2. \(\cos{(\frac{\pi}{2}-x)}=\sin{x}\)

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

    \(\cos{(\pi-x)}=-\cos{x}\)

    \(\cos{(\pi+x)}=-\cos{x}\)

Info

You can remember these formulas using a simple process:

  1. Check whether the angle before \(x\) is an odd multiple of \(\frac{\pi}{2}\), or a multiple of \(\pi\).
  2. If it is a multiple of \(\frac{\pi}{2}\), change the trigonometric ratio as follows: \(sin \leftrightarrow cos\), \(tan \leftrightarrow cot\), \(cosec \leftrightarrow sec\).
  3. Depending on which quadrant final angle is, find the sign of original trigonometric ratio in that quadrant. Add that as sign in final result.
  4. Finally replace the angle with just \(x\).

Example: For \(\tan{(\frac{3\pi}{2}+x)}\),

  1. Interchange \(tan\) with \(cot\) because \(\frac{3\pi}{2}\) is odd multiple of \(\frac{\pi}{2}\).
  2. Since \(\frac{3\pi}{2}+x\) is in \(4^{th}\) quadrant, \(tan\) is negative.

So, \(\tan{(\frac{3\pi}{2}+x)}=-\cot{x}\).

Trigonometric Ratios for Triple Angles

Check this short to understand: Trigonometric Ratios for Allied Angles

Double angles

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

You can express all the trigonometric ratios in terms of \(\tan{x}\). This is useful to solve equations containing double angles of multiple trigonometric ratios.

Trigonometric Ratios for Double Angles

Check this short to understand: Trigonometric Ratios for Double Angles

Triple angles

  1. \(\sin{3x}=3\sin{x}-4\sin^3{x}\)
  2. \(\cos{3x}=4\cos^3{x}-3\cos{x}\)
  3. \(\tan{3x}=\frac{3\tan{x}-\tan^3{x}}{1-3\tan^2{x}}\)
Trigonometric Ratios for Triple Angles

Check this short to understand: Trigonometric Ratios for Triple Angles

Transformation formulae

Sum & difference of trigonometric ratios

  1. \(\cos{x}+\cos{y}=2\cos{\frac{x+y}{2}}\cos{\frac{x-y}{2}}\)
  2. \(\cos{x}-\cos{y}=-2\sin{\frac{x+y}{2}}\sin{\frac{x-y}{2}}\)
  3. \(\sin{x}-\sin{y}=2\sin{\frac{x+y}{2}}\cos{\frac{x-y}{2}}\)
  4. \(\sin{x}-\sin{y}=2\cos{\frac{x+y}{2}}\sin{\frac{x-y}{2}}\)

Product of trigonometric ratios

  1. \(2\cos{x}\cos{y}=\cos{(x+y)}+\cos{(x-y)}\)
  2. \(-2\sin{x}\sin{y}=\cos{(x+y)}-\cos{(x-y)}\)
  3. \(2\sin{x}\cos{y}=\sin{(x+y)}+\sin{(x-y)}\)
  4. \(2\cos{x}\sin{y}=\sin{(x+y)}-\sin{(x-y)}\)
Note

You can derive the product of trigonometric rations formulas using the sum & difference of trigonometric ratios.

Transformation Formulae

Check this short to understand: Transformation Formulae