Question

What are the steps to solve trigonometric equations?

Answer 1

The steps to solve the trigonometric equations are:

1. Reduce the given trigonometric equation in terms of the basic trigonometric ratios.
2. Factorize the given trigonometric polynomial in terms of the ratio.
3. Solve each factor and write down the general solution.

Example:

${\tan }^{2}x+1=\frac{{\sin }^{2}x}{{\cos }^{2}x}+1$ $\left[\because \frac{\sin x}{\cos x}=\tan x\right]$

$\Rightarrow$ ${\tan }^{2}x+1=\frac{{\sin }^{2}x}{{\cos }^{2}x}+\frac{{\cos }^{2}x}{{\cos }^{2}x}$

$\Rightarrow$ ${\tan }^{2}x+1=\frac{{\sin }^{2}x+{\cos }^{2}x}{{\cos }^{2}x}$

$\Rightarrow$ ${\tan }^{2}x+1=\frac{1}{{\cos }^{2}x}$ $\left[\because {\sin }^{2}x+{\cos }^{2}x=1\right]$

$\Rightarrow$ ${\tan }^{2}x+1={\sec }^{2}x$ $\left[\because \frac{1}{\cos x}=\sec x\right]$