# Methods for Finding Principal Values of Trigonometric Equations

Introduction: The principal value of a trigonometric function is the value within a specified range that falls within the principal period of the function. Finding the principal value is essential for obtaining a unique solution to trigonometric equations. Here are methods for determining the principal value:

### 1. Understand Principal Periods:

• Principal Periods:
• Recognize the principal periods of common trigonometric functions:
• Sine ($\mathrm{sin}$) and cosine ($\mathrm{cos}$): $2\pi$
• Tangent ($\mathrm{tan}$): $\pi$
• Cotangent ($\mathrm{cot}$): $\pi$
• Secant ($\mathrm{sec}$) and cosecant ($\mathrm{csc}$): $2\pi$

### 2. Use Inverse Trigonometric Functions:

• Inverse Trigonometric Functions:

• When solving equations involving inverse trigonometric functions (e.g., ${\mathrm{sin}}^{-1}\left(x\right)$, ${\mathrm{cos}}^{-1}\left(x\right)$, ${\mathrm{tan}}^{-1}\left(x\right)$), use the properties of these functions to find the principal value.
• Example:

• For ${\mathrm{sin}}^{-1}\left(x\right)=\frac{\pi }{4}$, the principal value of $x$ is $\mathrm{sin}\left(\frac{\pi }{4}\right)$, which is $\frac{\sqrt{2}}{2}$

• Periodic Functions:

• If solving an equation involving a periodic function (e.g., $\mathrm{sin}\left(x\right)$, $\mathrm{cos}\left(x\right)$), consider adjusting the solution to obtain the principal value within the principal period.
• Example:

• For $\mathrm{cos}\left(x\right)=-\frac{1}{2}$, the general solutions are $x=\frac{2\pi }{3}+2n\pi$ and $x=\frac{4\pi }{3}+2n\pi$. The principal values within $\left[0,2\pi \right)$ are $x=\frac{2\pi }{3}$ and $x=\frac{4\pi }{3}$.

### 4. Restrict the Domain:

• Domain Restrictions:

• Be mindful of domain restrictions for inverse trigonometric functions. Adjust solutions to fit within the valid domain.
• Example:

• For ${\mathrm{tan}}^{-1}\left(x\right)=\frac{\pi }{4}$, the principal value is $x=\mathrm{tan}\left(\frac{\pi }{4}\right)$, which is $1$. However, consider the domain restriction $-\frac{\pi }{2}, so the principal value is $x=1$ within this range.

### 5. Graphical Approach:

• Graphical Visualization:

• Utilize graphing tools to visualize the graph of the trigonometric function and identify the principal values within the specified range.
• Example:

• Graph the equation and observe where the curve intersects the specified range to find principal values.

### 6. Check for Multiple Solutions:

• Multiple Solutions:
• Some trigonometric equations may have multiple solutions. Verify which solutions fall within the principal range to determine the principal value.

### 7. Consider Common Values:

• Common Trigonometric Values:
• Memorize common trigonometric values for special angles (${0}^{\circ }$, $3{0}^{\circ }$, $4{5}^{\circ }$, $6{0}^{\circ }$, $9{0}^{\circ }$) to quickly identify principal values.

### 8. Examples:

• $\mathrm{sin}\left(x\right)=\frac{\sqrt{3}}{2}$

• The principal value is $x=\frac{\pi }{3}$ within the range $\left[0,2\pi \right)$.
• ${\mathrm{cos}}^{-1}\left(x\right)=\frac{\pi }{4}$

• The principal value is $x={\mathrm{cos}}^{-1}\left(\frac{\sqrt{2}}{2}\right)$, which is $\frac{\pi }{4}$.