General Solution of Trigonometric Equations

Suppose a simple trigonometric equation \[\sin x=\frac{1}{2}\] This equation is satisfied by $x=30°$, $x=150°$ and all the other angles which differ from these by any integral multiple of $360°$. Thus, for all 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.

Any angle $x$ and each of the angles formed by adding or subtracting any integral multiple of $360°$ to or from the angle $x$, have the same initial and terminal arms, i.e. are coterminal. Therefore, any trigonometric function of an angle $x$ has the same value as the same trigonometric function of every angle coterminal with the angle.

In particular, \[\sin 30°=\sin(\pm 360°+30°)\]\[=\frac{1}{2}=\sin(n 360°+30°)\] where, $n=0,\pm 1,\pm 2, \pm 3,…$

The General Solution of sinx=k (-1≤k≤1)

Let $\theta$ be a particular angle such that $\sin\theta=k$. Then, the other value of $x$ for which $\sin x=k$, is $π-\theta$. Since the value of the trigonometric function is unaltered by adding or subtracting any integral multiple $360°$ to or from $\theta$, we have,

\[x=2mπ+\theta=2mπ+(-1)^{2m}\theta\text{ __(1)}\] and, \[x=2mπ+π-\theta\]\[\text{or,}\;x=(2m+1)π+(-1)^{2m+1} \theta\text{ __(2)}\]

where, $m$ is an integer, as the solution of the equation $\sin x=k$.

Combining $\text{(1)}$ and $\text{(2)}$, we have, \[x=nπ+(-1)^n\theta\] where, $n=0,\pm 1,\pm 2, \pm 3,…$

But $x=nπ+(-1)^n(\pm\theta)$ i.e. $x=nπ\pm(-1)^n\theta$ is same as to write $x=nπ\pm\theta$. Because if $n$ is even, $(-1)^n=1$, so \[x=nπ\pm(-1)^n\theta=nπ\pm\theta\] and if $n$ is odd, $(-1)^n=-1$, so \[x=nπ\pm(-1)^n\theta=nπ\pm\theta\]

$\therefore x=nπ\pm(-1)^n\theta=nπ\pm\theta$, for all integer values of $n$.

Cor. 1 If $k=0$, $\sin x=\sin nπ$, or $x=nπ$ for all integer values of $n$.

Cor. 2 If $\operatorname{cosec}x=k$, $x=nπ+(-1)^n\theta$, for all integer values of $n$.

The General Solution of cosx=k (-1≤k≤1)

Let $\theta$ be a particular angle such that $\cos\theta=k$. Then, the other values of $x$ for which $\cos x=k$, is $-\theta$. Since all the coterminal angles of $\theta$ and $-\theta$ are given by $2nπ+\theta$ and $2nπ-\theta$, \[\cos x=\cos(2nπ\pm\theta)\] \[\therefore x=2nπ\pm\theta\] for any integer values of $n$.

Cor. 1 If $k=1$, $\cos x=\cos 2nπ$ i.e. $x=2nπ$ for all integer values of $n$.

Cor. 2 If $\sec x=k$, then $x=2nπ\pm\theta$ for all integer values of $n$.

Cor. 3 If $k=0$, $\cos x=\cos\frac{π}{2}$ and $x=(2n+1)\frac{π}{2}$.

The General Solution of tanx=k

Let $\theta$ be a particular angle such that $\tan\theta=k$. Then, the other values of $x$ for which $\tan x=k$, is $π+\theta$. Thus, the coterminal angles are given by, \[2mπ+\theta\;\text{and}\;2mπ+π+\theta=(2m+1)π+\theta\] where, $m$ is an integer, as the solution of the equation $\tan x=k$.

\[\therefore\tan x=\tan(nπ+\theta)\] \[\therefore x=nπ+\theta\] for any integer values of $n$.

Cor.1 If $k=0$, $\tan x=\tan nπ$, or $x=nπ$ for any integer values of $n$.

Cor.2 If $\cot x=k$, then $x=nπ+\theta$ for any integer values of $n$.


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