CBSE Class 10 Maths Notes: Some Applications of Trigonometry

Angle of Elevation and Angle of Depression: Definitions & Diagram Conventions

Let’s dive into the fascinating world of how trigonometry helps us measure heights and distances! This chapter focuses on applying trigonometric ratios to solve real-world problems.

Definitions:

• Angle of Elevation: The angle formed when an observer looks upwards to an object. It is the angle between the horizontal line of sight and the line of sight to the object.
• Angle of Depression: The angle formed when an observer looks downwards to an object. It is the angle between the horizontal line of sight and the line of sight to the object.

Diagram Conventions:

To solve problems effectively, we use the following conventions:

• The observer’s eye is considered as a point.
• The object is also considered as a point (unless stated otherwise).
• We form right-angled triangles to apply trigonometric ratios.
• The horizontal line of sight is usually drawn from the observer’s eye.
• Always draw a clear diagram with labels for all known and unknown values.

Solving Height and Distance Problems

The core of this chapter involves using trigonometric ratios to calculate heights and distances.

Core Principles:

• Right-Angled Triangles: We primarily work with right-angled triangles.
• Trigonometric Ratios: Recall the basic trigonometric ratios:
• $sin(\theta) = \frac{Opposite}{Hypotenuse}$
• $cos(\theta) = \frac{Adjacent}{Hypotenuse}$
• $tan(\theta) = \frac{Opposite}{Adjacent}$
• Choosing the Correct Ratio: Select the trigonometric ratio that relates the known values (sides/angles) to the unknown value you want to find.
• Simple Problems: Problems are limited to involving at most two right triangles.

Important Reminders:

Draw a neat and labeled diagram for every problem, this will make problem-solving simple and clear.

Trigonometric Ratios for Standard Angles

We’ll mainly focus on problems involving the following standard angles:

Formulaes:

The values of trigonometric ratios for standard angles are crucial for solving problems. Memorize these!

• $sin(30°) = \frac{1}{2}$, $cos(30°) = \frac{\sqrt{3}}{2}$, $tan(30°) = \frac{1}{\sqrt{3}}$
• $sin(45°) = \frac{1}{\sqrt{2}}$, $cos(45°) = \frac{1}{\sqrt{2}}$, $tan(45°) = 1$
• $sin(60°) = \frac{\sqrt{3}}{2}$, $cos(60°) = \frac{1}{2}$, $tan(60°) = \sqrt{3}$

Example Problems

Let’s look at a couple of example problems to understand the approach:

Example 1:

A ladder leans against a wall. The angle of elevation is $60°$ and the foot of the ladder is 4.6 m away from the wall. Find the length of the ladder.

• Solution:
• Draw a right-angled triangle.
• Identify the adjacent side (distance from the wall) and the hypotenuse (ladder length).
• Use $cos(\theta) = \frac{Adjacent}{Hypotenuse}$
• $cos(60°) = \frac{4.6}{Hypotenuse}$
• $\frac{1}{2} = \frac{4.6}{Hypotenuse}$
• Hypotenuse = 9.2 m

Example 2:

The angle of depression of a car from a 75 m high cliff is $30°$. Find the distance of the car from the base of the cliff.

• Solution:
• Draw a right-angled triangle.
• Identify the opposite side (height of cliff) and the adjacent side (distance of car).
• Use $tan(\theta) = \frac{Opposite}{Adjacent}$
• $tan(30°) = \frac{75}{Adjacent}$
• $\frac{1}{\sqrt{3}} = \frac{75}{Adjacent}$
• Adjacent = $75\sqrt{3}$ m