Class 10 Maths Applications of Trigonometry Notes

9.1 Heights and Distances

In real-life problems, trigonometry can be applied to determine heights, distances, and angles of elevation and depression. The chapter “Applications of Trigonometry” primarily deals with these practical applications. One of the most common scenarios where trigonometry is used is in the calculation of heights and distances, such as finding the height of a building, a tree, or the distance between two objects when only the angles and distances are known.

Key Concepts in Heights and Distances

To understand heights and distances, we use the basic trigonometric ratios (sine, cosine, and tangent) in conjunction with the concept of angles of elevation and angles of depression.

• Angle of Elevation: This is the angle formed by the line of sight from an observer to an object above the observer’s eye level.
• Angle of Depression: This is the angle formed by the line of sight from an observer to an object below the observer’s eye level.

Real-life Example of Heights and Distances

Imagine you’re standing at a certain point and looking up at the top of a tall building. The angle at which you’re looking up is called the angle of elevation. Similarly, if you’re looking down at an object from a certain height, the angle at which you’re looking down is called the angle of depression.

Let’s now look at the mathematical application of trigonometry for calculating heights and distances using these concepts.

---

Application 1: Finding the Height of an Object

When you know the distance from a point to the base of an object (like a tree or a building) and the angle of elevation to the top of the object, you can use trigonometric ratios to find the height of the object.

Consider a scenario:

• The distance between a point and the base of the building is 30 meters.
• The angle of elevation from the point to the top of the building is $45^\circ$45∘.

We can use the tangent ratio to solve for the height of the building. The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side (height) to the adjacent side (distance from the building):$$\tan \theta = \frac{\text{Height}}{\text{Distance from the base}}$$tanθ=Distance from the baseHeight​

For this example:$$\tan 45^\circ = \frac{\text{Height}}{30}$$tan45∘=30Height​

Since $\tan 45^\circ = 1$tan45∘=1, the equation simplifies to:$$1 = \frac{\text{Height}}{30}$$1=30Height​

So, the height of the building is:$$\text{Height} = 30 \, \text{meters}$$Height=30meters

---

Application 2: Finding the Distance Between Two Points

Let’s consider another scenario where you are standing at a point, looking at the top of a building, and you know the height of the building and the angle of elevation.

Suppose the height of the building is 40 meters and the angle of elevation is $30^\circ$30∘. To find the distance between you and the base of the building, we can use the tan function again.

Using the formula:$$\tan \theta = \frac{\text{Height}}{\text{Distance}}$$tanθ=DistanceHeight​

We can rearrange it to solve for the distance:$$\text{Distance} = \frac{\text{Height}}{\tan \theta}$$Distance=tanθHeight​

Substitute the given values:$$\text{Distance} = \frac{40}{\tan 30^\circ}$$Distance=tan30∘40​

Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$tan30∘=3​1​, we get:$$\text{Distance} = \frac{40}{\frac{1}{\sqrt{3}}} = 40 \times \sqrt{3} \approx 69.28 \, \text{meters}$$Distance=3​1​40​=40×3​≈69.28meters

So, the distance from you to the building is approximately 69.28 meters.

---

Application 3: Using Angles of Depression

Angles of depression are used to calculate distances and heights when the observer is at a higher point, such as from the top of a tower or a hill, looking down at an object. The principle is similar to that of the angle of elevation, but the observer’s line of sight is directed downward.

For example, consider a person on top of a tower. The person looks down at a car parked on the ground. If the height of the tower is known and the angle of depression is given, trigonometric functions can be used to find the distance between the car and the base of the tower.

---

Key Trigonometric Ratios for Heights and Distances

Here are the essential trigonometric ratios used in the applications of heights and distances:

• Tangent: $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$tanθ=AdjacentOpposite​
• Sine: $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$sinθ=HypotenuseOpposite​
• Cosine: $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$cosθ=HypotenuseAdjacent​

The appropriate ratio is chosen based on the given information (angle and sides of the triangle).

---

9.2 Summary

In this chapter, we studied the practical applications of trigonometry in solving problems related to heights and distances. Key concepts included:

• Angles of Elevation and Depression: These angles help determine the relationship between distances and heights in real-world scenarios.
• Trigonometric Ratios: The sine, cosine, and tangent ratios are applied to find heights and distances based on known angles.
• Practical Problems: We used these concepts to solve problems involving the calculation of the height of objects (like buildings or trees) and the distance between objects.

Trigonometry has widespread applications in various fields, such as navigation, architecture, astronomy, and surveying. By mastering these applications, students can solve real-life problems involving angles and distances.

MCQs Based on the “Applications of Trigonometry” Chapter:

1. If the height of a building is 20 meters and the angle of elevation from a point 15 meters away from its base is 30∘30^\circ30∘, what is the height of the building?

a) 15 meters
b) 20 meters
c) 25 meters
d) 30 meters

Answer: b) 20 meters

2. A person standing 50 meters away from a tower observes the top of the tower at an angle of elevation of 60∘60^\circ60∘. What is the height of the tower?

a) 50 meters
b) 60 meters
c) 86.6 meters
d) 100 meters

Answer: c) 86.6 meters

3. The angle of depression from the top of a tower is 45∘45^\circ45∘ to a point on the ground. If the distance between the point and the base of the tower is 40 meters, what is the height of the tower?

a) 40 meters
b) 20 meters
c) 30 meters
d) 60 meters

Answer: a) 40 meters