JEE Main & Advanced Mathematics Trigonometric Equations Trigonometrical Equations and Inequations Definition

An equation involving one or more trigonometrical ratio of an unknown angle is called a trigonometrical equation

i.e., \[\sin x+\cos 2x=1\],\[(1-\tan \theta )(1+\sin 2\theta )=1+\tan \theta \], \[|\sec \left( \theta +\frac{\pi }{4} \right)|\text{ }=2\] etc.

A trigonometric equation is different from a trigonometrical identities. An identity is satisfied for every value of the unknown angle e.g.,\[{{\cos }^{2}}x=1-{{\sin }^{2}}x\]is true \[\forall x\in R\], while a trigonometric equation is satisfied for some particular values of the unknown angle.

(1) Roots of trigonometrical equation : The value of unknown angle (a variable quantity) which satisfies the given equation is called the root of an equation, e.g., \[\cos \theta =\frac{1}{2}\], the root is \[\theta ={{60}^{o}}\] or \[\theta ={{300}^{o}}\] because the equation is satisfied if we put \[\theta ={{60}^{o}}\]or \[\theta ={{300}^{o}}\].

(2) Solution of trigonometrical equations : A value of the unknown angle which satisfies the trigonometrical equation is called its solution.

Since all trigonometrical ratios are periodic in nature, generally a trigonometrical equation has more than one solution or an infinite number of solutions. There are basically three types of solutions:

(i) Particular solution : A specific value of unknown angle satisfying the equation.

(ii) Principal solution : Smallest numerical value of the unknown angle satisfying the equation (Numerically smallest particular solution).

(iii) General solution : Complete set of values of the unknown angle satisfying the equation. It contains all particular solutions as well as principal solutions.