Trigonometric Equations and Inequalities
1.0 Definition
1.0 Definition
"Trigonometry" means measuring the sides of a triangle. As we all know where every triangle comes, we have to deal not only with the sides but also with the angles of a triangle. We have already studied in earlier classes the relation between sides and angles of a triangle and its use in trigonometric ratios. In this chapter, we are going to study the application of trigonometric ratios in terms of trigonometric equations and inequalities.
- The Equations containing trigonometric functions of unknown angles are known as Trigonometric Equations. Our focus in this section is to find out the solutions of those equations containing unknown angle.
Illustration 1. $$\sin \theta = \frac{1}{2}$$$$\cos \theta + \sin \theta = \frac{1}{{\sqrt 2 }}$$$$\cos \theta = \frac{1}{{\sqrt 2 }}$$
Solution of Trigonometric Equations is the value of the unknown angles that satisfy the equations.
Illustration 2.
$$\begin{equation} \begin{aligned} \sin \theta = \frac{1}{2} \\ \Rightarrow \theta = \frac{\pi }{6},\pi - \frac{\pi }{6},2\pi + \frac{\pi }{6},3\pi - \frac{\pi }{6}.....so\;on.\quad \quad \left...( 1 \right) \\ \Rightarrow \theta = n\pi + {\left( { - 1} \right)^n}\frac{\pi }{6},\quad n \in Z\quad \quad \quad \left...( 2 \right) \\\end{aligned} \end{equation} $$
NOTE:
- Since the Trigonometric functions are periodic in nature, they have infinitely many solutions, as in (1) and are called Principal Solution.
- In (2), the solution is generalized by means of periodicity and is known as General Solution.
