# Trigonometrical Equations

Trigonometric equations are mathematical equations that involve trigonometric functions and unknown angles. These equations express relationships between the angles of a triangle and the ratios of its sides, as described by trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant.

The general form of a trigonometric equation is:

$f\left(\theta \right)=g\left(\theta \right)$

Here, $f\left(\theta \right)$ and $g\left(\theta \right)$ are expressions involving trigonometric functions, and $\theta$ is the unknown angle. Solving a trigonometric equation involves finding the values of $\theta$ that satisfy the equation.

Trigonometric equations can take various forms, including linear, quadratic, cubic, or involving multiple angles. They may also include inverse trigonometric functions. The solutions to these equations are often expressed in terms of general solutions, considering the periodic nature of trigonometric functions.

Trigonometric equations involve trigonometric functions and unknown angles. These equations express relationships between the angles of a triangle and the ratios of its sides, as described by trigonometric functions such as sine ($\mathrm{sin}$), cosine ($\mathrm{cos}$), tangent ($\mathrm{tan}$), cotangent ($\mathrm{cot}$), secant ($\mathrm{sec}$), and cosecant ($\mathrm{csc}$).

### Types of Trigonometric Equations:

1. Linear Trigonometric Equations:

• Example: $\mathrm{sin}\left(x\right)=\frac{1}{2}$
• Solutions: $x=\frac{\pi }{6}+2n\pi$ and $x=\frac{5\pi }{6}+2n\pi$, 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: ${\mathrm{sin}}^{-1}\left(x\right)=\frac{\pi }{4}$
• Solution: $x=\mathrm{sin}\left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}$
4. Multiple-Angle Trigonometric Equations:

• Example: $\mathrm{tan}\left(2x\right)+1=0$
• Solutions: $x=\frac{\pi }{8}+\frac{n\pi }{2}$
5. Trigonometric Equations with Fractions:

• Example: $\mathrm{cos}\left(\frac{x}{2}\right)=-\frac{1}{\sqrt{2}}$
• Solutions: $x=\frac{3\pi }{2}+4n\pi$

### Steps for Solving Trigonometric Equations:

1. Isolate the Trigonometric Function:

• Move all terms involving the trigonometric function to one side of the equation.
2. Apply Trigonometric Identities:

• Utilize trigonometric identities to simplify the equation.
3. Use Algebraic Techniques:

• Employ algebraic techniques such as factoring, completing the square, or substitution.
4. Consider the Periodicity:

• Remember the periodic nature of trigonometric functions and find general solutions.
5. Verify Solutions:

• Check obtained solutions by substituting them back into the original equation.
6. Express General Solutions:

• Present solutions in general form considering periodic repetitions.