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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
α-\alphaα-\alphasin(α)-sin(\alpha)cos(α)cos(\alpha)tg(α)-tg(\alpha)ctg(α)-ctg(\alpha)
90+α90^\circ + \alphaπ2+α\frac{\pi}{2} + \alphacos(α)cos(\alpha)sin(α)-sin(\alpha)ctg(α)-ctg(\alpha)tg(α)-tg(\alpha)
90α90^\circ - \alphaπ2α\frac{\pi}{2} - \alphacos(α)cos(\alpha)sin(α)sin(\alpha)ctg(α)ctg(\alpha)tg(α)tg(\alpha)
180+α180^\circ + \alphaπ+α\pi + \alphasin(α)-sin(\alpha)cos(α)-cos(\alpha)tg(α)tg(\alpha)ctg(α)ctg(\alpha)
180α180^\circ - \alphaπα\pi - \alphasin(α)sin(\alpha)cos(α)-cos(\alpha)tg(α)-tg(\alpha)ctg(α)-ctg(\alpha)
270+α270^\circ + \alpha32π+α\frac{3}{2}\pi + \alphacos(α)-cos(\alpha)sin(α)sin(\alpha)ctg(α)-ctg(\alpha)tg(α)-tg(\alpha)
270α270^\circ - \alpha32πα\frac{3}{2}\pi - \alphacos(α)-cos(\alpha)sin(α)-sin(\alpha)ctg(α)ctg(\alpha)tg(α)tg(\alpha)
360+α360^\circ + \alpha2π+α2\pi + \alphasin(α)sin(\alpha)cos(α)cos(\alpha)tg(α)tg(\alpha)ctg(α)ctg(\alpha)
360α360^\circ - \alpha2πα2\pi - \alphasin(α)-sin(\alpha)cos(α)cos(\alpha)tg(α)-tg(\alpha)tg(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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