Some Applications of Trig..

Learning Objectives

• Understand and calculate the angles of elevation and depression using trigonometric ratios in right triangles.
• Apply trigonometric ratios such as tangent, sine, and cosine to solve problems involving right triangles, including finding unknown heights and distances.
• Solve real-world problems involving heights and distances using trigonometric principles, often requiring the calculation of angles of elevation and depression.
• Use the tangent function to calculate unknown heights in problems involving angles of elevation and depression.

Chapter Notes

Angle of Elevation and Depression

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

Trigonometric Ratios in Right Triangles

• Tangent Function: $\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}$
• Sine Function: $\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$
• Cosine Function: $\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$

Height and Distance Problems

• Problem Solving Steps:
  1. Identify the right triangle in the problem.
  2. Determine which trigonometric ratio to use based on the given angles and sides.
  3. Use the appropriate trigonometric ratio to find the unknown side or angle.

Use of Tangent in Height Calculations

• Example: To find the height of a tower:
  • Given: Distance from the tower base (e.g., 15 m) and angle of elevation (e.g., 60°).
  • Use: $\tan 60° = \frac{\text{Height of Tower}}{\text{Distance from Tower}}$
  • Solve for the height of the tower.

Example Problems

• Example 1: A tower stands vertically on the ground. From a point 15 m away, the angle of elevation is 60°. Find the height of the tower.
  • Solution: $\tan 60° = \frac{\text{Height of Tower}}{15} \Rightarrow \text{Height} = 15\sqrt{3} \text{ m}$
• Example 2: An electrician needs to reach a point 1.3 m below the top of a 5 m pole with a ladder inclined at 60°.
  • Solution: Use $\sin 60°$ to find the ladder length and $\cot 60°$ to find the distance from the pole.
• Example 3: From a point 28.5 m away, the angle of elevation to the top of a chimney is 45°.
  • Solution: $\tan 45° = \frac{\text{Height of Chimney}}{28.5} \Rightarrow \text{Height} = 30 \text{ m}$
• Example 4: From a point, the angle of elevation to the top of a 10 m building is 30°, and to the top of a flagstaff on the building is 45°.
  • Solution: Solve using $\tan 30°$ and $\tan 45°$ to find the flagstaff height and distance from the building.
• Example 5: A tower's shadow is 40 m longer when the sun's altitude is 30° than when it is 60°.
  • Solution: Use $\tan 60°$ and $\tan 30°$ to find the tower height.
• Example 6: The angles of depression from a multi-storeyed building to the top and bottom of an 8 m building are 30° and 45°.
  • Solution: Use trigonometric ratios to find the height of the multi-storeyed building and the distance between buildings.
• Example 7: From a bridge, angles of depression to river banks are 30° and 45°.
  • Solution: Use $\tan 30°$ and $\tan 45°$ to find the river width.