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)
  • ⓘ Hint: If you are interested in trigonometry you can checkout our other calculators:
    • reduction formulas - so-called reduction formulas table, that help to calculate value of trigonometric functions for less common angles,
    • trigonometric functions values - a table containing the values of trigonometric functions for the most common angles, e.g. sin 90 degrees,
    • trigonometric identities - a list of different, more or less popular, dependencies between various trigonometric functions.

How to use this tool#

  • 1. First, convert your angle into one of below forms:
    • π2±α\dfrac{\pi}{2} \pm \alpha,
    • π±α\pi \pm \alpha,
    • 32π±α\dfrac{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) = \dfrac{\sqrt{3}}{2}
    That's all folks!

Tags and links to this website#

What tags this calculator has#


This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

Links to external sites (leaving Calculla?)#

JavaScript failed !
So this is static version of this website.
This website works a lot better in JavaScript enabled browser.
Please enable JavaScript.