# General solution of standard trigonometrical equations

Introduction: The general solution of a trigonometric equation refers to an expression that represents all possible solutions by incorporating an integer parameter. This concept is particularly important for equations involving periodic functions such as sine, cosine, tangent, cotangent, secant, and cosecant.

### 1. Periodicity of Trigonometric Functions:

• Principal Periods:
• Sine and cosine have a principal period of $2\pi$, while tangent, cotangent, secant, and cosecant have a principal period of $\pi$.
• This means the functions repeat their values every $2\pi$ or $\pi$ radians.

### 2. General Solution Formulas:

• Linear Trigonometric Equations:

• $\mathrm{sin}\left(\theta \right)=a$or $\mathrm{cos}\left(\theta \right)=a$ has the general solution $\theta ={\mathrm{sin}}^{-1}\left(a\right)+2n\pi$ or $\theta ={\mathrm{cos}}^{-1}\left(a\right)+2n\pi$, where $n$ is an integer.

• ${\mathrm{sin}}^{2}\left(\theta \right)=a$ or ${\mathrm{cos}}^{2}\left(\theta \right)=a$ has the general solution $\theta ={\mathrm{sin}}^{-1}\left(\sqrt{a}\right)+2n\pi$ or $\theta ={\mathrm{cos}}^{-1}\left(\sqrt{a}\right)+2n\pi$, where $n$ is an integer.
• Tangent and Cotangent Equations:

• $\mathrm{tan}\left(\theta \right)=a$ or $\mathrm{cot}\left(\theta \right)=a$ has the general solution $\theta ={\mathrm{tan}}^{-1}\left(a\right)+n\pi$ or $\theta ={\mathrm{cot}}^{-1}\left(a\right)+n\pi$, where $n$ is an integer.
• Secant and Cosecant Equations:

• $\mathrm{sec}\left(\theta \right)=a$ or $\mathrm{csc}\left(\theta \right)=a$ has the general solution $\theta ={\mathrm{sec}}^{-1}\left(a\right)+n\pi$ or $\theta ={\mathrm{csc}}^{-1}\left(a\right)+n\pi$, where $n$ is an integer.

• Due to the periodic nature of trigonometric functions, solutions may repeat at intervals of $2\pi$ or $\pi$. Adjust the general solution accordingly by adding $2n\pi$ or $n\pi$ to represent all possible solutions.

### 4. Verification:

• Check Solutions:
• Verify obtained solutions by substituting them back into the original equation. Ensure they satisfy the equation within the given range.

### 5. Examples:

• Linear Equation:

• Solve $\mathrm{sin}\left(\theta \right)=\frac{1}{2}$. The general solution is $\theta ={\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)+2n\pi$.
• Solve ${\mathrm{cos}}^{2}\left(\theta \right)=\frac{3}{4}$. The general solution is $\theta ={\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)+2n\pi$.
• Solve $\mathrm{tan}\left(\theta \right)=-1$. The general solution is $\theta ={\mathrm{tan}}^{-1}\left(-1\right)+n\pi$.