# 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:

1. Linear Equations:

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

• Example: ${\mathrm{cos}}^{2}\left(x\right)-\mathrm{cos}\left(x\right)=0$
• Solutions: $x=0,\frac{2\pi }{3},\pi ,\frac{4\pi }{3}$, and so on.
3. Inverse Trigonometric Equations:

• Example: ${\mathrm{tan}}^{-1}\left(x\right)=\frac{\pi }{4}$
• 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.