Some Applications of Trigonometry Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Introduction to Trigonometry Applications

• Trigonometry is widely used to solve problems involving heights and distances. This involves calculating unknown heights, distances, or angles using trigonometric ratios in right-angled triangles.

2. Important Terms and Definitions

• Line of Sight: The straight line drawn from the observer’s eye to the point being viewed.
• Angle of Elevation: The angle between the horizontal line (from the observer's eye) and the line of sight to a point above the horizontal level.
• Angle of Depression: The angle between the horizontal line (from the observer's eye) and the line of sight to a point below the horizontal level.

3. Trigonometric Ratios Recap

• Sine (sin θ): The ratio of the length of the opposite side to the hypotenuse. $$\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$$
• Cosine (cos θ): The ratio of the length of the adjacent side to the hypotenuse. $$\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$$
• Tangent (tan θ): The ratio of the length of the opposite side to the adjacent side. $$\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}$$

4. Angle of Elevation

• When an observer looks up at an object, the angle formed between the horizontal line of sight and the line of sight to the object is the angle of elevation.
• Example: Calculating the height of a building using the angle of elevation and the distance from the building.

5. Angle of Depression

• When an observer looks down at an object, the angle formed between the horizontal line of sight and the line of sight to the object is the angle of depression.
• Example: Calculating the distance of a boat from the top of a lighthouse using the angle of depression.

6. Solving Problems Involving Heights and Distances

• To solve problems, the following steps are generally followed:
  1. Identify the right-angled triangle in the problem.
  2. Identify the angle of elevation or depression.
  3. Use the appropriate trigonometric ratio based on the given information (sine, cosine, or tangent).
  4. Set up the equation and solve for the unknown value (height, distance, or angle).

7. Common Real-Life Applications

• Determining the height of a tower or a tree using the angle of elevation.
• Finding the distance between two ships using the angle of depression from a lighthouse.
• Calculating the height of a mountain using the angles of elevation from two different points.

Formula Sheet

1. Trigonometric Ratios in a Right-Angled Triangle:

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

2. Angle of Elevation:

   $$\tan \theta = \frac{\text{Height of Object}}{\text{Distance from Object}}$$

   (When the observer's eye level is at the base of the object.)

3. Angle of Depression:

   $$\tan \theta = \frac{\text{Height of Observer Above the Object}}{\text{Horizontal Distance to the Object}}$$

   (The angle of depression is equal to the angle of elevation from the object to the observer.)

4. Height of an Object Using Trigonometry:

   $$\text{Height} = \text{Distance} \times \tan \theta$$

   (This formula is used when the distance from the object and the angle of elevation are known.)

5. Distance from Object Using Trigonometry:

   $$\text{Distance} = \frac{\text{Height}}{\tan \theta}$$

   (This formula is used when the height of the object and the angle of elevation are known.)

Examples for Practice

1. A man standing 50 meters away from a tower observes the top of the tower at an angle of elevation of 30°. Find the height of the tower.
2. A ladder is leaning against a wall, making an angle of 60° with the ground. If the ladder is 10 meters long, find the height at which the ladder touches the wall.
3. From the top of a lighthouse 100 meters high, the angle of depression of a boat is 45°. Find the distance of the boat from the base of the lighthouse.