The sine function is a fundamental element in trigonometry often denoted as \( \sin \theta \). It links a right triangle's angle to the ratio of the lengths of its sides. The formula \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) illustrates its simplicity and power.
To utilize the sine function in calculations:
- Identify the opposite side relative to the angle \( \theta \): for our problem, it's the elevation gain (160 m).
- Identify the hypotenuse: the total inclined path (2400 m).
- Substitute these values to form the ratio.
Once the ratio is determined, the next step involves figuring out \( \theta \) using the inverse sine or arcsine function. Specifically, you compute \( \arcsin \left( \frac{1}{15} \right) \), yielding an angle of about 3.82 degrees. Applying these calculations helps in understanding the trigonometric functions' applications in real-world scenarios, like road elevation.