Chapter 13

Chapter 13

Chapter Overview

This chapter deals with the concept of Trigonometry, which is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Trigonometry is an essential tool for solving problems in various fields such as physics, engineering, navigation, and more. In this chapter, we will explore the fundamental concepts of trigonometry, including the definitions of trigonometric ratios, the use of trigonometric identities, and the application of trigonometry to solve problems.

Learning Objectives

• Understand the definitions of trigonometric ratios
• Learn to use trigonometric identities to simplify expressions
• Apply trigonometry to solve problems in various fields
• Understand the concept of periodicity and symmetry in trigonometric functions
• Learn to use trigonometric functions to model real-world phenomena

Important Concepts

Introduction to Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is an essential tool for solving problems in various fields such as physics, engineering, navigation, and more. Trigonometry has its roots in ancient civilizations, with the Egyptians and Babylonians using trigonometric methods to build pyramids and calculate astronomical positions. In modern times, trigonometry is used in a wide range of applications, from satellite navigation to medical imaging.

Trigonometric Ratios

The trigonometric ratios are defined as the ratios of the lengths of the sides of a right-angled triangle. The six basic trigonometric ratios are:

• Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. For example, in a right-angled triangle with a hypotenuse of 10 cm and an opposite side of 6 cm, the sine of the angle opposite the 6 cm side is 0.6.
• Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. For example, in a right-angled triangle with a hypotenuse of 10 cm and an adjacent side of 8 cm, the cosine of the angle adjacent to the 8 cm side is 0.8.
• Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. For example, in a right-angled triangle with an opposite side of 6 cm and an adjacent side of 3 cm, the tangent of the angle opposite the 6 cm side is 2.
• Cosecant (csc): The ratio of the length of the hypotenuse to the length of the opposite side. For example, in a right-angled triangle with a hypotenuse of 10 cm and an opposite side of 6 cm, the cosecant of the angle opposite the 6 cm side is 1.67.
• Secant (sec): The ratio of the length of the hypotenuse to the length of the adjacent side. For example, in a right-angled triangle with a hypotenuse of 10 cm and an adjacent side of 8 cm, the secant of the angle adjacent to the 8 cm side is 1.25.
• Cotangent (cot): The ratio of the length of the adjacent side to the length of the opposite side. For example, in a right-angled triangle with an adjacent side of 3 cm and an opposite side of 6 cm, the cotangent of the angle opposite the 6 cm side is 0.5.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the trigonometric functions. Some of the basic trigonometric identities are:

• Pythagorean identity: sin^2(x) + cos^2(x) = 1. This identity is true for all values of x and can be used to simplify expressions involving sine and cosine.
• Complementary angle identity: sin(x) = cos(90 - x). This identity states that the sine of an angle is equal to the cosine of its complementary angle.
• Supplementary angle identity: sin(x) = -cos(180 - x). This identity states that the sine of an angle is equal to the negative of the cosine of its supplementary angle.

Periodicity and Symmetry

Trigonometric functions have periodicity and symmetry properties. The sine and cosine functions have a period of 360° or 2π radians, while the tangent function has a period of 180° or π radians. This means that the values of these functions repeat every 360° or 2π radians.

Graphs of Trigonometric Functions

The graphs of trigonometric functions are periodic and have symmetry properties. The sine and cosine functions have a maximum value of 1 and a minimum value of -1, while the tangent function has a maximum value of infinity and a minimum value of -infinity.

Advanced Section: Deep-Dive Case Studies and Real-Life Applications

Case Study 1: Navigation

Trigonometry is used in navigation to determine distances and directions. For example, a pilot uses trigonometry to calculate the distance to a nearby airport and the direction to fly in order to reach it.

Case Study 2: Physics

Trigonometry is used in physics to describe the motion of objects. For example, the trajectory of a projectile can be modeled using trigonometric functions.

Case Study 3: Engineering

Trigonometry is used in engineering to design and build structures. For example, the design of a bridge requires the use of trigonometric functions to calculate the stress and strain on the structure.

Case Study 4: Computer Science

Trigonometry is used in computer science to create 3D graphics and simulations. For example, the movement of a 3D object in a video game can be modeled using trigonometric functions.

Advanced Section: Step-by-Step Problem Solving Strategies & Detailed Proofs

Step-by-Step Problem Solving Strategy 1: Using Trigonometric Identities

1. Identify the trigonometric identity that can be used to simplify the expression.
2. Apply the identity to simplify the expression.
3. Repeat steps 1 and 2 until the expression is simplified.

Step-by-Step Problem Solving Strategy 2: Using Trigonometric Ratios

1. Identify the trigonometric ratio that can be used to solve the problem.
2. Apply the ratio to solve the problem.
3. Repeat steps 1 and 2 until the problem is solved.

Detailed Proof 1: Pythagorean Identity

sin^2(x) + cos^2(x) = 1

Proof:

Let x be an angle in a right-angled triangle with a hypotenuse of length 1.

Using the Pythagorean theorem, we know that the sum of the squares of the lengths of the legs of the triangle is equal to the square of the length of the hypotenuse.

Let the length of the leg opposite the angle x be a, and the length of the leg adjacent to the angle x be b.

Then, we have:

a^2 + b^2 = 1

Now, let's consider the trigonometric ratio sin(x) = a/1.

We can rewrite this as:

sin^2(x) = a^2/1^2

Using the Pythagorean theorem, we know that a^2 + b^2 = 1.

Therefore, we can rewrite the expression as:

sin^2(x) = (a^2 + b^2)/1^2

Simplifying, we get:

sin^2(x) = 1

Now, let's consider the trigonometric ratio cos(x) = b/1.

We can rewrite this as:

cos^2(x) = b^2/1^2

Using the Pythagorean theorem, we know that a^2 + b^2 = 1.

Therefore, we can rewrite the expression as:

cos^2(x) = (1 - a^2)/1^2

Simplifying, we get:

cos^2(x) = 1 - sin^2(x)

Now, we can substitute this expression into the previous equation:

sin^2(x) + cos^2(x) = sin^2(x) + (1 - sin^2(x))

Simplifying, we get:

sin^2(x) + cos^2(x) = 1

Therefore, we have proven the Pythagorean identity.

Advanced Section: Higher-Order Thinking Skills (HOTS) Questions

Question 1: Using Trigonometric Identities

If sin(x) = 3/5, find cos(x).

Question 2: Using Trigonometric Ratios

If a right-angled triangle has a hypotenuse of length 10 cm and an opposite side of length 6 cm, find the length of the adjacent side.

Question 3: Applying Trigonometry to Real-World Problems

A pilot is flying a plane at an altitude of 5000 feet. The pilot wants to know the distance to a nearby airport that is located at a bearing of 270°. Use trigonometry to find the distance to the airport.

Advanced Section: Previous Year Questions (PYQs) with Solutions

Question 1: Using Trigonometric Identities (2019)

If sin(x) = 2/3, find cos(x).

Solution:

Using the Pythagorean identity, we know that sin^2(x) + cos^2(x) = 1.

We can rewrite this as:

(2/3)^2 + cos^2(x) = 1

Simplifying, we get:

4/9 + cos^2(x) = 1

Subtracting 4/9 from both sides, we get:

cos^2(x) = 5/9

Taking the square root of both sides, we get:

cos(x) = ±√(5/9)

Since cos(x) is positive, we can take the positive square root:

cos(x) = √(5/9)

Question 2: Using Trigonometric Ratios (2018)

If a right-angled triangle has a hypotenuse of length 12 cm and an opposite side of length 9 cm, find the length of the adjacent side.

Solution:

Using the Pythagorean theorem, we know that the sum of the squares of the lengths of the legs of the triangle is equal to the square of the length of the hypotenuse.

Let the length of the leg adjacent to the angle x be a.

Then, we have:

a^2 + 9^2 = 12^2

Simplifying, we get:

a^2 + 81 = 144

Subtracting 81 from both sides, we get:

a^2 = 63

Taking the square root of both sides, we get:

a = √63

Advanced Section: NCERT Textbook Questions & Detailed Answers

Question 1: Using Trigonometric Identities (NCERT Textbook Question 1)

If sin(x) = 3/5, find cos(x).

Solution:

Using the Pythagorean identity, we know that sin^2(x) + cos^2(x) = 1.

We can rewrite this as:

(3/5)^2 + cos^2(x) = 1

Simplifying, we get:

9/25 + cos^2(x) = 1

Subtracting 9/25 from both sides, we get:

cos^2(x) = 16/25

Taking the square root of both sides, we get:

cos(x) = ±√(16/25)

Since cos(x) is positive, we can take the positive square root:

cos(x) = √(16/25)

Question 2: Using Trigonometric Ratios (NCERT Textbook Question 2)

If a right-angled triangle