General Solutions of Simple Trigonometric Equations
Definition :
An equation involving trigonometric functions is called a trigonometric equation
Example :-\(
\sin x + \cos x = 0,\cos ^2 x - \sin x = \frac{1}
{4}
\) , \(
2\cos ^2 x + 3\sin ^2 x = 0,a\cos \theta + b\sin \theta = c
\) etc. are all trigonometric equations
The number ‘\(
\alpha
\)’ which satisfies a trigonometric equation is called the solution set (or) the general solution of the equation
Consider the equation \(
\sin \theta = \frac{1}
{2}
\) for \(
\sin \theta = \frac{1}
{2}
\), we have \(
\sin \frac{\pi }
{6} = \frac{1}
{2}
\).
Therefore \(
\frac{\pi }
{6}
\) is a solution of the equation \(
\sin \theta = \frac{1}
{2}
\).
Also \(
\sin (\pi - \frac{\pi }
{6}) = \sin \frac{{5\pi }}
{6} = \frac{1}
{2}[\because \sin (\pi - \theta ) = \sin \theta ]\frac{{5\pi }}
{6}
\) is also a solution of .
Therefore all the angles coterminons with and also satisfy the equation .
\(
\therefore
\) Therefore all the angles coterminons with \(
\frac{{\pi }}
{6}
\) and \(
\frac{{5\pi }}
{6}
\)
also satisfy the equation \(
\sin \theta = \frac{1}
{2}
\).
Thus a trigonometric equation has infinitely many solutions.
Note :
(1) An Algebraic equation will have as many solutions as the degree of equation (i.e., finite in number), where as Trigonometric equation will have infinitely many solutions.
(2) We don’t define the degree of a Trigonometric equation.