Some Application of Trigonometry Class 10 ||Maths|| Chapter 9 NCERT Notes

Trigonometry, the study of the relationships between the sides and angles of triangles, is very useful in solving real-life problems involving heights and distances. This chapter primarily deals with the use of trigonometric ratios to determine unknown distances and heights in practical situations.

Key Concepts:

Line of Sight:
The line drawn from an observer's eye to the point being viewed is called the line of sight. When the object is above the horizontal level of the observer, the line of sight is inclined upwards. Similarly, if the object is below, it is inclined downwards.
Horizontal Line:
A horizontal line is a straight line parallel to the ground and perpendicular to the vertical direction.
Angle of Elevation:
The angle formed between the horizontal line and the line of sight when an observer looks up at an object is called the angle of elevation.
- Example: If you are looking at the top of a building from the ground, the angle between your line of sight and the horizontal ground is the angle of elevation.
Angle of Depression:
The angle formed between the horizontal line and the line of sight when an observer looks down at an object is called the angle of depression.
- Example: If you are standing on top of a cliff and looking down at a boat in the water, the angle between your line of sight and the horizontal line (parallel to the ground) is the angle of depression.

Trigonometric Ratios and Their Applications:

To solve problems involving heights and distances, the basic trigonometric ratios are applied. The primary trigonometric ratios are:

Sine (sin):
$\sin (θ) = \frac{Opposite side}{Hypotenuse}$
Cosine (cos):
$\cos (θ) = \frac{Adjacent side}{Hypotenuse}$
Tangent (tan):
$\tan (θ) = \frac{Opposite side}{Adjacent side}$

These ratios are used to determine unknown sides or angles in a right-angled triangle.

Important Considerations:

Right-Angled Triangles:
In practical situations, the height and distance problems are modeled using right-angled triangles. One side often represents the height of an object (like a building or a tree), another side represents the distance from the object, and the angle is either the angle of elevation or depression.
Measuring Heights:
If you know the distance from the object and the angle of elevation, you can find the height using the trigonometric ratios.
Measuring Distances:
If you know the height of an object and the angle of elevation, you can find the horizontal distance between the observer and the object.

Step-by-Step Process to Solve Problems:

Read and Understand the Problem:
Determine if you are dealing with an angle of elevation or angle of depression and identify what values are given (e.g., height, distance, or angle).
Draw a Diagram:
Sketch the situation described in the problem. Label the right-angled triangle formed by the observer's line of sight and the object.
Identify Known Values:
Write down the values provided in the problem, such as the height, angle of elevation, or depression, and distance from the object.
Choose the Appropriate Trigonometric Ratio:
Based on the given values and the unknown value you need to find, decide which trigonometric ratio to use (sine, cosine, or tangent).
Set Up the Equation:
Using the trigonometric ratio, set up an equation and solve for the unknown value (height, distance, or angle).
Solve and Interpret:
Once you solve the equation, interpret the result in the context of the problem.

Example Problems:

Example 1: Finding the Height of a Tree Using Angle of Elevation

An observer is standing 50 meters away from the base of a tree. The angle of elevation to the top of the tree is 30°. Find the height of the tree.

Solution:

Draw the triangle:
The distance from the observer to the tree is the adjacent side, and the height of the tree is the opposite side.
Identify the trigonometric ratio:
Since we know the angle (30°) and the adjacent side (50 m), we can use the tan(θ) ratio:
$\tan (30 °) = \frac{Opposite}{Adjacent} = \frac{Height of the tree}{50}$
Solve for the height:
$\tan (30 °) = \frac{1}{\sqrt{3}}$
So,
$\frac{1}{\sqrt{3}} = \frac{h}{50}$
Multiply both sides by 50 to get the height (h):
$h = \frac{50}{\sqrt{3}} \approx 28.87 meters$
Therefore, the height of the tree is approximately 28.87 meters.

Example 2: Finding the Distance Using Angle of Depression

From the top of a 100-meter high cliff, the angle of depression to a boat is 45°. Find the distance between the base of the cliff and the boat.

Solution:

Draw the triangle:
The height of the cliff is the opposite side, and the distance from the cliff to the boat is the adjacent side. The angle of depression is 45°.
Identify the trigonometric ratio:
Since we know the height (100 m) and the angle (45°), we use the tan(θ) ratio:
$\tan (45 °) = \frac{Opposite}{Adjacent} = \frac{100}{Distance}$
Solve for the distance:
$\tan (45 °) = 1$
So,
$1 = \frac{100}{Distance}$
Therefore, the distance = 100 meters.

Thus, the boat is 100 meters away from the base of the cliff.

Important Points to Remember:

Always identify whether you are using the angle of elevation or the angle of depression in a given problem.
Draw diagrams to visualize the problem clearly and to help with setting up the correct trigonometric ratios.
In right-angled triangles, you can apply Pythagoras' Theorem if necessary to find unknown sides if two sides are known.
Use the standard trigonometric values for commonly used angles like 30°, 45°, and 60°.

This chapter focuses on the practical use of trigonometric ratios in real-life situations like calculating heights of buildings, trees, towers, or distances between objects. Understanding how to apply these concepts is crucial for solving height and distance problems.

If you'd like further explanations or need help with specific problems, just let me know! 😊

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