- Define the tangent function as the ratio of the opposite side to the adjacent side in a right triangle.
- Recognize that \(\tan(22.6^{\circ}) \approx \frac{5}{12}\), which relates to a 5-12-13 triangle.
- Identify that the equation \(\tan(22.6^{\circ}) = \frac{a}{12}\) is the only one that correctly fits the tangent ratio definition, treating \(a\) as the opposite side.
- Conclude that the correct equation is \(\boxed{\tan(22.6^{\circ})=\frac{a}{12}}\).
Explanation
- Understanding the Tangent Function in a Right Triangle
The tangent of an angle \(\theta\) in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]
The problem provides the angle \(\theta = 22.6^{\circ}\). A well-known right triangle involving the numbers 5, 12, and 13 has a tangent value of approximately \(\tan(22.6^{\circ}) \approx \frac{5}{12}\). Given this, we can evaluate the provided equations to see which one follows the standard definition of the tangent function.
- Evaluating the Options
We have the following candidate equations:
- \(\tan(22.6^{\circ}) = \frac{12}{a}\)
- \(\tan(22.6^{\circ}) = \frac{a}{13}\)
- \(\tan(22.6^{\circ}) = \frac{a}{12}\)
- \(\tan(22.6^{\circ}) = \frac{13}{a}\)
Given the approximation \(\tan(22.6^{\circ}) \approx \frac{5}{12}\), comparing this to the options, equation (3) \(\tan(22.6^{\circ}) = \frac{a}{12}\) aligns with the definition \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\), where \(a\) is the opposite side (with \(a \approx 5\)) and 12 is the adjacent side. The other options involve ratios like \(\frac{12}{a}\) (cotangent), \(\frac{a}{13}\) (sine), or \(\frac{13}{a}\) (cosecant), which do not represent the tangent function.
- Conclusion
Based on the analysis, the equation that correctly represents the relationship for \(\tan(22.6^{\circ})\) in a right triangle where 12 is the adjacent side is \(\tan(22.6^{\circ}) = \frac{a}{12}\). This matches the standard trigonometric ratio for tangent.
The final answer is \(\boxed{\tan(22.6^{\circ})=\frac{a}{12}}\).
Examples
Imagine you are working as an architect designing a roof slope. To ensure the drainage is perfect, you need a roof pitch of approximately 22.6 degrees. If you know the horizontal distance (the run) of your roof is 12 meters, you can use the tangent function to find the required vertical height (the rise) of the roof. By setting up the equation \(\tan(22.6^{\circ}) = \frac{\text{rise}}{12}\), you calculate that the rise must be approximately \(12 \times 0.416 \approx 5 \text{ meters}\). This simple calculation helps ensure the architectural design is both precise and functional.