What is Some Applications of Trigonometry Class 10: Key Concepts Explained

Understanding Some Applications of Trigonometry in Class 10

Trigonometry studies the relationships between angles and sides of triangles. In Class 10 NCERT Mathematics, "Some Applications of Trigonometry" focuses on using these relationships to solve practical problems, especially those involving right-angled triangles. Students learn to apply trigonometric ratios—sine ($\sin$), cosine ($\cos$), and tangent ($\tan$)—to find unknown lengths or angles.

This chapter is important because it connects abstract mathematical concepts to real-world situations, such as measuring inaccessible heights or distances without direct measurement. It builds on the basics of trigonometry introduced earlier and emphasizes problem-solving skills essential for exams and daily life.

Key Trigonometric Ratios and Their Role in Applications

The three primary trigonometric ratios used in Class 10 applications are:

• $\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$
• $\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$
• $\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$

These ratios help relate an angle in a right-angled triangle to the lengths of its sides. For example, if you know an angle and one side, you can find the other sides using these ratios.

Worked Example:

A ladder leans against a wall, making an angle of 60° with the ground. If the ladder is 10 m long, find the height it reaches on the wall.

Using $\sin 60^\circ = \frac{\text{Height}}{10}$,

$$\text{Height} = 10 \times \sin 60^\circ = 10 \times \frac{\sqrt{3}}{2} = 8.66 \text{ m}$$

This shows how trigonometric ratios solve height problems efficiently.

Common Real-Life Problems Solved Using Trigonometry

Some Applications of Trigonometry in Class 10 include:

• Finding the height of a building or tree: By measuring the angle of elevation from a point and the distance from the base.
• Determining the distance across a river: Using angles and trigonometric ratios without crossing the river.
• Calculating the length of a shadow: When the height and angle of elevation of the sun are known.

These problems often involve right-angled triangles where one side or angle is unknown. Trigonometry provides a systematic way to find these unknowns.

Example: If the angle of elevation of the top of a tower from a point 30 m away is 45°, the height of the tower is:

Using $\tan 45^\circ = \frac{\text{Height}}{30}$,

$$\text{Height} = 30 \times 1 = 30 \text{ m}$$

Angles of Elevation and Depression: Definitions and Uses

In the chapter, two important concepts are:

• Angle of Elevation: The angle between the horizontal line and the line of sight when looking upward.
• Angle of Depression: The angle between the horizontal line and the line of sight when looking downward.

These angles help in solving problems related to heights and distances. For example, when you look at the top of a tower from a distance, the angle your line of sight makes with the horizontal is the angle of elevation.

Example: A person standing 50 m from a building observes the top at an angle of elevation of 30°. The building's height can be found using:

$$\tan 30^\circ = \frac{\text{Height}}{50} \Rightarrow \text{Height} = 50 \times \frac{1}{\sqrt{3}} = 28.87 \text{ m}$$

Step-by-Step Approach to Solving Trigonometry Application Problems

To solve problems in Some Applications of Trigonometry, follow these steps:

1. Draw a diagram: Visualize the problem with all known and unknown quantities. 2. Identify the right triangle: Mark the angle of elevation or depression. 3. Label sides: Opposite, adjacent, and hypotenuse relative to the angle. 4. Choose the correct trigonometric ratio: Based on known and unknown sides. 5. Set up the equation: Using $\sin$, $\cos$, or $\tan$. 6. Solve for the unknown: Calculate the required length or angle.

This method ensures clarity and accuracy in solving exam problems.

Comparing Trigonometric Ratios for Different Angles

Understanding how trigonometric ratios change with angles is crucial. Here's a comparison of $\sin$, $\cos$, and $\tan$ values for some common angles:

This table helps quickly estimate values during problem-solving and understand the behaviour of these ratios as angles increase.