Trigonometric Equation

An equation involving one or more trigonometric functions of a variable is called a trigonometric equation. The equation may be true for one or more values, but not for every value of the variable. The values of the variable (angle) are known as the root(s) of the equation.

The simplest type of trigonometric equation is that in which a trigonometric function of a variable is a constant. For example, $\sin x=\frac{1}{2}$ is satisfied by $x=30°$, $x=150°$ and all the other angles which differ from these by any integral multiple of $360°$. Hence, for all the integers, the solutions are \[x=30°+n\cdot 360°\;\text{and}\;x=150°+n\cdot 360°\]

Thus, there exists an infinite number of roots of the equation $\sin x=\frac{1}{2}$.

The set of all possible solutions of a trigonometric equation form the general solution of the equation.

Trigonometric Equation in Other Forms

If a given trigonometric equation is not in the simplest form, then we can solve it by deriving two or more simple equations which will yield all the solutions of the given equation. In order to obtain simple equations from a given equation, the most useful tools are the algebraic operations and trigonometric identities.

While solving trigonometric equations, algebraic operations such as squaring, cubing, etc. may give rise to some additional equations and consequently some additional roots. It is therefore advisable to check which of the roots thus obtained do not satisfy the given equation. Such roots must be discarded.

For example: Consider an equation, \[\cos x-\sin x=1\;\;\text{for}\;0≤x≤360°\]

Now, let us solve this equation and find the roots which will satisfy the given equation. \[\cos x=1+\sin x\] Squaring both sides, we get, \[\cos^2x=1+2\sin x+\sin^2x\]

\[\text{or,}\;1-\sin^2x=1+2\sin x+\sin^2x\] \[\text{or,}\;2\sin^2x+2\sin x=0\] \[\text{or,}\;\sin x(\sin x+1)=0\] \[\text{Either,}\;\;\sin x=0\;\;\text{or}\;\;\sin x=-1\]

Thus, if $\sin x=0$, then $x=0°$ and $x=180°$ in the given range. If $\sin x=-1$, then $x=270°$ in the same range.

Here, $x=0°$ and $x=270°$ satisfy the given equation $\cos x-\sin x=1$ whereas the value $x=180°$ does not. Hence, $x=180°$ should be discarded.