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Table of trigonometric reduction formulas
Table shows so-called rediction formulas, which allow to calculate values of trigonometric functions of obtuse angle (more than 90 degrees) without calculator easily.

Trigonometric reduction formulas

Angle in degreesAngle in radiansSineCosineTangensCotangens
Show sourceα-\alphaShow sourceα-\alphaShow sourcesin(α)-sin(\alpha)Show sourcecos(α)cos(\alpha)Show sourcetg(α)-tg(\alpha)Show sourcectg(α)-ctg(\alpha)
Show source90+α90^\circ + \alphaShow sourceπ2+α\frac{\pi}{2} + \alphaShow sourcecos(α)cos(\alpha)Show sourcesin(α)-sin(\alpha)Show sourcectg(α)-ctg(\alpha)Show sourcetg(α)-tg(\alpha)
Show source90α90^\circ - \alphaShow sourceπ2α\frac{\pi}{2} - \alphaShow sourcecos(α)cos(\alpha)Show sourcesin(α)sin(\alpha)Show sourcectg(α)ctg(\alpha)Show sourcetg(α)tg(\alpha)
Show source180+α180^\circ + \alphaShow sourceπ+α\pi + \alphaShow sourcesin(α)-sin(\alpha)Show sourcecos(α)-cos(\alpha)Show sourcetg(α)tg(\alpha)Show sourcectg(α)ctg(\alpha)
Show source180α180^\circ - \alphaShow sourceπα\pi - \alphaShow sourcesin(α)sin(\alpha)Show sourcecos(α)-cos(\alpha)Show sourcetg(α)-tg(\alpha)Show sourcectg(α)-ctg(\alpha)
Show source270+α270^\circ + \alphaShow source32π+α\frac{3}{2}\pi + \alphaShow sourcecos(α)-cos(\alpha)Show sourcesin(α)sin(\alpha)Show sourcectg(α)-ctg(\alpha)Show sourcetg(α)-tg(\alpha)
Show source270α270^\circ - \alphaShow source32πα\frac{3}{2}\pi - \alphaShow sourcecos(α)-cos(\alpha)Show sourcesin(α)-sin(\alpha)Show sourcectg(α)ctg(\alpha)Show sourcetg(α)tg(\alpha)
Show source360+α360^\circ + \alphaShow source2π+α2\pi + \alphaShow sourcesin(α)sin(\alpha)Show sourcecos(α)cos(\alpha)Show sourcetg(α)tg(\alpha)Show sourcectg(α)ctg(\alpha)
Show source360α360^\circ - \alphaShow source2πα2\pi - \alphaShow sourcesin(α)-sin(\alpha)Show sourcecos(α)cos(\alpha)Show sourcetg(α)-tg(\alpha)Show sourcetg(α)-tg(\alpha)

Some facts

  • Reduction formulas allow conversion of trigonometric expressions of obtuse angle into equivalent (and simpler) form containing acute angle.
  • We often prefer expression containing obtuse angle, because math tables contain values of trigonometric functions for these angles.
  • The basis of all reduction formulas are fact, that trigonometric functions are periodic. It means that their values cyclically repeat every certain angle. This angle is called period.
    ⓘ Example: The basic period of sine function is 2π2\pi (360360^\circ), because:
    sin(α+2π)=sin(α)sin(\alpha + 2\pi) = sin(\alpha)
    ⓘ Example: The basic period of tangens function is π\pi (180180^\circ), because:
    tg(α+π)=tg(α)tg(\alpha + \pi) = tg(\alpha)

How to use this tool

  • 1. First, convert your angle into one of below forms:
    • π2±α\frac{\pi}{2} \pm \alpha,
    • π±α\pi \pm \alpha,
    • 32π±α\frac{3}{2}\pi \pm \alpha,
    • 2π±α2\pi \pm \alpha.
    Or using degrees:
    • 90±α90^\circ \pm \alpha,
    • 180±α180^\circ \pm \alpha,
    • 270±α270^\circ \pm \alpha,
    • 360±α360^\circ \pm \alpha.
  • 2. Next, find row containing your new angle in reduction formulas table.
  • 3. Finally, find column containing your trigonometric function and replace your expression with one from table.
  • ⓘ Example: We want to calculate sine of 120 degrees.
    • 1. We found, that 120 degrees can be written as:
      120=90+30120^\circ = 90^\circ + 30^\circ
    • 2. We see that our new angle matches to below form in reduction formulas table: 90+α90^\circ + \alpha
    • 3. We look at sine column and we finally found:
      sin(90+30)=cos(30)=32sin(90^\circ + 30^\circ) = cos(30^\circ) = \frac{\sqrt{3}}{2}
    That's all folks!

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