Trigonometric equation

A trigonometric equation (also known as a goniometric equation ) is an equation in which the variable to be determined occurs in the argument of trigonometric functions ( trigonometric functions ). In solving these equations, the relationship between the trigonometric functions is helpful, especially the addition theorems.

Number of solutions

Because of the periodicity of the trigonometric functions, trigonometric equations generally have an infinite number of solutions . By restricting the basic set to a " basic interval" (for example [ 0,2 · π ] or [ 0, π ]) one reduces the number of solutions to a finite number or one describes the solutions by a periodicity sum (like k · 2 · π or k π ).

example

The trigonometric equation

${\displaystyle \sin \;x=\cos \;x}$

one can transform using the relationship to ${\displaystyle \cos x={\sqrt {1-\sin ^{2}\;x}}}$

${\displaystyle \sin \;x={\sqrt {1-\sin ^{2}\;x}}.}$

Squaring gives

${\displaystyle \sin ^{2}\;x=1-\sin ^{2}\;x}$

and it

${\displaystyle 2\cdot \sin ^{2}\;x=1,}$

so

${\displaystyle \sin \;x=\pm \;{\sqrt {\frac {1}{2}}}}$

with the solutions

${\displaystyle x=45^{\circ }\pm k\cdot 90^{\circ }\quad (k=0,1,2,...)}$

or in radians

${\displaystyle x={\frac {\pi }{4}}\pm k\cdot {\frac {\pi }{2}}\qquad (k=0,1,2,...).}$

Since squaring is not an equivalence transformation , these solutions must be verified against the output equation . This gives valid solutions to the initial equation

${\displaystyle x={\frac {\pi }{4}}\pm 2k\cdot {\frac {\pi }{2}}\qquad (k=0,1,2,...).}$

Individual evidence

1. ^ Arnfried Kemnitz: Mathematics at the beginning of the course . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-1741-9 , pp. 75 .