Trigonometric ratios for angles of elevation and depression problems.

📚 Trigonometric Ratios: Angles of Elevation and Depression

Trigonometry is used to find unknown lengths or angles in right-angled triangles. When dealing with real-world scenarios, we often encounter angles of elevation and depression. These angles help us relate the sides of the triangle to the angle formed with the horizontal.

📜 History and Background

The principles of trigonometry date back to ancient civilizations, including the Egyptians, Babylonians, and Greeks. Early applications were primarily in astronomy and navigation. The systematic study of trigonometric functions and their relationships emerged over centuries, with mathematicians like Hipparchus and Ptolemy making significant contributions. The concepts of angles of elevation and depression became particularly useful in surveying and military applications.

📌 Key Principles

• 📐 Angle of Elevation: The angle formed by the line of sight and the horizontal plane when looking upwards. Imagine you are standing on the ground looking up at the top of a building. The angle between your eye-level (horizontal) and your line of sight to the top of the building is the angle of elevation.
• 📉 Angle of Depression: The angle formed by the line of sight and the horizontal plane when looking downwards. Think of standing on top of a cliff and looking down at a boat in the sea. The angle between your eye-level (horizontal) and your line of sight to the boat is the angle of depression.
• ➗ Trigonometric Ratios: The primary trigonometric ratios are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). These ratios relate the angles of a right triangle to the lengths of its sides:
  • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

🌍 Real-World Examples

Example 1: Finding the Height of a Building

A person stands 50 meters away from the base of a building. The angle of elevation to the top of the building is 60 degrees. Find the height of the building.

1. 📐 Identify the knowns:
   • Adjacent side = 50 m
   • Angle of elevation = 60°
2. ✍️ Choose the appropriate trigonometric ratio:

   We use the tangent function because we have the adjacent side and want to find the opposite side (height):

   $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

3. ➮ Set up the equation:

   $\tan(60^\circ) = \frac{\text{height}}{50}$
4. ➗ Solve for the height:

   $\text{height} = 50 \cdot \tan(60^\circ)$

   $\text{height} = 50 \cdot \sqrt{3}$

   $\text{height} \approx 50 \cdot 1.732 \approx 86.6$ meters

Example 2: Finding the Distance to a Boat

A person stands on top of a cliff that is 100 meters high. The angle of depression to a boat is 30 degrees. Find the horizontal distance from the base of the cliff to the boat.

1. 📐 Identify the knowns:
   • Opposite side (height of cliff) = 100 m
   • Angle of depression = 30°
2. ✍️ Choose the appropriate trigonometric ratio:

   We use the tangent function because we have the opposite side and want to find the adjacent side (horizontal distance):

   $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

3. ➮ Set up the equation:

   $\tan(30^\circ) = \frac{100}{\text{distance}}$

4. ➗ Solve for the distance:

   $\text{distance} = \frac{100}{\tan(30^\circ)}$

   $\text{distance} = \frac{100}{\frac{1}{\sqrt{3}}}$

   $\text{distance} = 100 \cdot \sqrt{3} \approx 173.2$ meters

💡 Conclusion

Understanding trigonometric ratios in the context of angles of elevation and depression is crucial for solving various real-world problems. By correctly identifying the angles and applying the appropriate trigonometric functions, we can determine unknown lengths and distances effectively.