Summary of Applications of Trigonometry

Heights and Distances

• Trigonometric Ratios: Used to determine heights and distances in real-life scenarios.
• Line of Sight: The line drawn from the observer's eye to the object viewed.
• Angle of Elevation: The angle formed by the line of sight with the horizontal when the object is above the observer's eye level.
• Angle of Depression: The angle formed by the line of sight with the horizontal when the object is below the observer's eye level.

Key Examples

• Example 1: Height of a tower can be calculated using the angle of elevation from a point on the ground.
• Example 3: Height of a chimney determined using the angle of elevation from an observer's eye level.
• Example 4: Length of a flagstaff and distance from a point to a building calculated using angles of elevation.
• Example 6: Height of a multi-storeyed building and distance between buildings found using angles of depression.
• Example 7: Width of a river calculated using angles of depression from a bridge.

Important Formulas

• Height Calculation:
  • For angle of elevation:
    • \( an( ext{angle}) = rac{ ext{height}}{ ext{distance}}
  • For angle of depression:
    • \( an( ext{angle}) = rac{ ext{height}}{ ext{distance}}

Common Mistakes & Exam Tips

• Mistake: Confusing angle of elevation with angle of depression.
  • Tip: Always identify the position of the observer and the object.
• Mistake: Incorrectly applying trigonometric ratios.
  • Tip: Ensure the correct triangle is used for calculations.
• Mistake: Forgetting to add the observer's height when calculating total height.
  • Tip: Always account for the height of the observer in height calculations.

Some Applications of Trigonometry

9.1 Heights and Distances

Key Concepts

• Line of Sight: The line drawn from the eye of an observer to the point in the object viewed by the observer.
• Angle of Elevation: The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.
• Angle of Depression: The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.

Examples

• Example 3: An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. The height of the chimney is calculated as follows:
  • Height of chimney = (28.5 + 1.5) m = 30 m.
• Example 4: From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°. The length of the flagstaff and the distance of the building from point P can be calculated using trigonometric ratios.

Important Formulas

• Tangent Ratio: For angle of elevation,
  • tan(θ) = opposite/adjacent
• Height Calculation:
  • Height = Distance × tan(Angle of Elevation)

Exercises

1. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°.
2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Common Mistakes and Exam Tips

Common Pitfalls

• Misidentifying Angles: Students often confuse angles of elevation and depression. Remember:
  • Angle of Elevation: When looking up at an object.
  • Angle of Depression: When looking down at an object.
• Incorrect Use of Trigonometric Ratios: Ensure you are using the correct ratio based on the triangle formed. For example:
  • Use tan for opposite/adjacent relationships.
  • Use sin for opposite/hypotenuse relationships.
  • Use cos for adjacent/hypotenuse relationships.
• Neglecting Units: Always pay attention to the units of measurement. Ensure consistency throughout your calculations.
• Forgetting to Add Heights: When calculating total heights, remember to add the height of the observer if applicable.

Tips for Success

• Draw Diagrams: Visual representation can help clarify the relationships between angles and distances.
• Label Everything: Clearly label all points, angles, and distances in your diagrams to avoid confusion.
• Practice with Examples: Work through various examples to familiarize yourself with different scenarios and applications of trigonometry.
• Check Your Work: Always review your calculations and ensure that your final answers make sense in the context of the problem.