Roots of trigonometrical equation
Roots of trigonometric equations are the values of the unknown angle that satisfy the given trigonometric equation. These equations involve trigonometric functions like sine ($\mathrm{sin}$), cosine ($\mathrm{cos}$), tangent ($\mathrm{tan}$), cotangent ($\mathrm{cot}$), secant ($\mathrm{sec}$), and cosecant ($\mathrm{csc}$). Understanding how to find and interpret these roots is crucial in solving problems related to angles and periodic functions.
1. Basic Concepts:

Definition:
 A root of a trigonometric equation is an angle for which the equation is satisfied.

Periodicity:
 Trigonometric functions are periodic, repeating their values at regular intervals. Consider the principal period of the function when finding roots.
2. Finding Roots:

Isolating the Trigonometric Function:
 Start by isolating the trigonometric function on one side of the equation.

Applying Trigonometric Identities:
 Use trigonometric identities to simplify the equation or express one trigonometric function in terms of another.

Algebraic Techniques:
 Employ algebraic techniques such as factoring, completing the square, or using quadratic formulas to simplify the equation.

Multiple Roots:
 Consider multiple roots within the given range, especially for periodic functions.
3. Common Trigonometric Equations:

Linear Equations:
 Example: $\mathrm{sin}(x)=0$
 Solutions: $x=n\pi $, where $n$ is an integer.

Quadratic Equations:
 Example: ${\mathrm{cos}}^{2}(x)\mathrm{cos}(x)=0$
 Solutions: $x=0,\frac{2\pi}{3},\pi ,\frac{4\pi}{3}$, and so on.

Inverse Trigonometric Equations:
 Example: ${\mathrm{tan}}^{1}(x)=\frac{\pi}{4}$
 Solutions: $x=\mathrm{tan}\left(\frac{\pi}{4}\right)=1$
4. Periodicity and General Solutions:

Periodicity of Trigonometric Functions:
 Sine and cosine have a principal period of $2\pi $, while tangent, cotangent, secant, and cosecant have a principal period of $\pi $.

General Solutions:
 Express the roots in terms of general solutions considering the periodic nature of trigonometric functions.