# General Solution of Specific Trigonometric Equations

Introduction: The general solution of trigonometric equations provides a comprehensive expression for all possible solutions, considering the periodic nature of trigonometric functions. This is particularly relevant for equations involving inverse trigonometric functions, multiple angles, and special values.

### 1. Inverse Trigonometric Equations:

• Example:

• Consider the equation ${\mathrm{sin}}^{-1}\left(x\right)=\frac{\pi }{4}$.
• General Solution:

• The general solution for ${\mathrm{sin}}^{-1}\left(x\right)=\frac{\pi }{4}$ is $x=\mathrm{sin}\left(\frac{\pi }{4}\right)+2n\pi$, where $n$ is an integer.
• This accounts for the periodic nature of the sine function.

### 2. Trigonometric Equations with Multiple Angles:

• Example:

• Solve $\mathrm{cos}\left(2x\right)+\mathrm{cos}\left(x\right)=1$.
• General Solution:

• The general solution for $\mathrm{cos}\left(2x\right)+\mathrm{cos}\left(x\right)=1$ involves expressing $x$ in terms of a multiple angle: $x=\frac{\pi }{3}+n\pi$

### 3. Trigonometric Equations with Special Values:

• Example:

• Solve $\mathrm{tan}\left(x\right)=-1$.
• General Solution:

• The general solution for $\mathrm{tan}\left(x\right)=-1$ is $x={\mathrm{tan}}^{-1}\left(-1\right)+n\pi$.
• This takes into account the periodicity of the tangent function.

### 4. Periodicity Consideration:

• Principal Periods:

• Be aware of the principal periods of trigonometric functions. Sine and cosine have a principal period of $2\pi$, while tangent has a principal period of $\pi$.

• Make adjustments to the general solution by adding $2n\pi$or $n\pi$ to represent all possible solutions.

### 5. Verification:

• Check Solutions:
• Always verify obtained solutions by substituting them back into the original equation to ensure consistency.

### 6. Examples:

• Inverse Trigonometric Equation:

• Solve ${\mathrm{cos}}^{-1}\left(x\right)=\frac{\pi }{3}$. The general solution is $x=\mathrm{cos}\left(\frac{\pi }{3}\right)+2n\pi$.
• Equation with Multiple Angles:

• Solve $\mathrm{sin}\left(2x\right)=\mathrm{cos}\left(x\right)$. The general solution is $x=\frac{\pi }{4}+n\pi$.
• Equation with Special Values:

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