Chapter 9

Chapter 9: Trigonometry

Chapter Overview

This chapter focuses on the concept of Trigonometry, which is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The chapter will cover the basic concepts of trigonometry, including the definition of trigonometric functions, their properties, and identities. It will also cover the concept of triangles, including right-angled triangles, and the use of trigonometric functions to solve problems related to these triangles.

Learning Objectives

• Understand the definition and properties of trigonometric functions
• Learn to solve problems related to right-angled triangles using trigonometric functions
• Understand the concept of identities and how to use them to simplify trigonometric expressions
• Apply trigonometric functions to solve problems in real-life situations

Important Concepts

Introduction to Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The word "trigonometry" comes from the Greek words "trigono" meaning triangle and "metron" meaning measure. Trigonometry is used to solve problems related to triangles, including right-angled triangles, and is used in many real-life situations such as navigation, physics, and engineering.

Trigonometry has a rich history, dating back to ancient civilizations. The ancient Greeks, such as Hipparchus and Euclid, made significant contributions to the field of trigonometry. They developed the concept of trigonometric functions and used them to solve problems related to triangles. The development of trigonometry continued through the centuries, with major contributions from mathematicians such as Pierre de Fermat and Isaac Newton.

Trigonometric Functions

Trigonometric functions are used to describe the relationships between the sides and angles of triangles. The six basic trigonometric functions are:

• Sine (sin): defined as the ratio of the length of the opposite side to the length of the hypotenuse
• Cosine (cos): defined as the ratio of the length of the adjacent side to the length of the hypotenuse
• Tangent (tan): defined as the ratio of the length of the opposite side to the length of the adjacent side
• Cosecant (csc): defined as the ratio of the length of the hypotenuse to the length of the opposite side
• Secant (sec): defined as the ratio of the length of the hypotenuse to the length of the adjacent side
• Cotangent (cot): defined as the ratio of the length of the adjacent side to the length of the opposite side

Trigonometric functions are used in a wide range of applications, including navigation, physics, and engineering. They are used to describe the relationships between the sides and angles of triangles, and are used to solve problems related to these triangles.

Properties of Trigonometric Functions

The trigonometric functions have several properties, including:

• Periodicity: the trigonometric functions have a periodic nature, meaning that they repeat themselves after a certain interval
• Symmetry: the trigonometric functions have symmetry about the origin, meaning that they are equal to their negative counterparts
• Identities: the trigonometric functions have several identities, such as the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1

The periodicity of trigonometric functions is an important property that is used in many applications. The periodicity of the sine and cosine functions, for example, is used to describe the motion of objects in physics. The symmetry of trigonometric functions is also an important property that is used in many applications. The symmetry of the sine and cosine functions, for example, is used to describe the relationships between the sides and angles of triangles.

Identities

The trigonometric functions have several identities, including:

• Pythagorean identity: sin^2(x) + cos^2(x) = 1
• Quotient identity: tan(x) = sin(x) / cos(x)
• Reciprocal identity: csc(x) = 1 / sin(x), sec(x) = 1 / cos(x), cot(x) = 1 / tan(x)

The Pythagorean identity is a fundamental identity in trigonometry that is used to describe the relationships between the sides and angles of triangles. The quotient identity is used to describe the relationships between the sine and cosine functions, and the reciprocal identity is used to describe the relationships between the trigonometric functions.

Key Definitions

• Trigonometric function: a function that describes the relationship between the sides and angles of a triangle
• Sine: the ratio of the length of the opposite side to the length of the hypotenuse
• Cosine: the ratio of the length of the adjacent side to the length of the hypotenuse
• Tangent: the ratio of the length of the opposite side to the length of the adjacent side
• Cosecant: the ratio of the length of the hypotenuse to the length of the opposite side
• Secant: the ratio of the length of the hypotenuse to the length of the adjacent side
• Cotangent: the ratio of the length of the adjacent side to the length of the opposite side

Important Terms

Term	Meaning
Trigonometry	the branch of mathematics that deals with the relationships between the sides and angles of triangles
Right-angled triangle	a triangle with one right angle (90 degrees)
Hypotenuse	the longest side of a right-angled triangle, opposite the right angle
Opposite side	the side of a right-angled triangle that is opposite the angle being considered
Adjacent side	the side of a right-angled triangle that is adjacent to the angle being considered

Important Formulas

• Pythagorean identity: sin^2(x) + cos^2(x) = 1
• Quotient identity: tan(x) = sin(x) / cos(x)
• Reciprocal identity: csc(x) = 1 / sin(x), sec(x) = 1 / cos(x), cot(x) = 1 / tan(x)

Diagrams (Description Only)

The chapter includes several diagrams and figures to illustrate the concepts of trigonometry, including:

• A right-angled triangle with the hypotenuse, opposite side, and adjacent side labeled
• A graph of the sine function
• A graph of the cosine function
• A graph of the tangent function

Real-Life Applications

Trigonometry has many real-life applications, including:

• Navigation: trigonometry is used in navigation to determine distances and directions
• Physics: trigonometry is used in physics to describe the motion of objects
• Engineering: trigonometry is used in engineering to design and build structures

Deep-Dive Case Studies and Real-Life Applications

Navigation

Trigonometry is used in navigation to determine distances and directions. For example, a navigator can use trigonometry to determine the distance between two points on a map. The navigator can use the sine and cosine functions to calculate the distance and direction between the two points.

Physics

Trigonometry is used in physics to describe the motion of objects. For example, a physicist can use trigonometry to describe the motion of a projectile. The physicist can use the sine and cosine functions to calculate the trajectory of the projectile and determine its speed and direction.

Engineering

Trigonometry is used in engineering to design and build structures. For example, an engineer can use trigonometry to design a bridge. The engineer can use the sine and cosine functions to calculate the stress and strain on the bridge and determine its stability.

Step-by-Step Problem Solving Strategies & Detailed Proofs

Step-by-Step Problem Solving Strategy

1. Identify the problem and determine the type of trigonometric function that is needed to solve it.
2. Use the trigonometric function to calculate the value of the unknown quantity.
3. Use the calculated value to determine the solution to the problem.

Detailed Proof of the Pythagorean Identity

The Pythagorean identity is a fundamental identity in trigonometry that is used to describe the relationships between the sides and angles of triangles. The identity is:

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

To prove this identity, we can start by using the definition of the sine and cosine functions:

sin(x) = opposite side / hypotenuse cos(x) = adjacent side / hypotenuse

We can then square both sides of these equations to get:

sin^2(x) = (opposite side / hypotenuse)^2 cos^2(x) = (adjacent side / hypotenuse)^2

We can then add these two equations together to get:

sin^2(x) + cos^2(x) = (opposite side / hypotenuse)^2 + (adjacent side / hypotenuse)^2

We can then simplify this equation by combining the two fractions:

sin^2(x) + cos^2(x) = (opposite side^2 + adjacent side^2) / hypotenuse^2

We can then use the Pythagorean theorem to simplify this equation further:

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

This proves the Pythagorean identity.

Higher-Order Thinking Skills (HOTS) Questions

Question 1

A right-angled triangle has a hypotenuse of length 10 cm and an adjacent side of length 6 cm. Find the length of the opposite side using the sine function.

Question 2

A projectile is launched at an angle of 30 degrees to the horizontal. Find the range of the projectile using the tangent function.

Question 3

A bridge has a length of 100 m and a height of 20 m. Find the angle of the bridge using the cosine function.

Previous Year Questions (PYQs) with solutions

Question 1

A right-angled triangle has a hypotenuse of length 10 cm and an adjacent side of length 6 cm. Find the length of the opposite side using the sine function.

Solution: Using the sine function, we can calculate the length of the opposite side as follows:

sin(x) = opposite side / hypotenuse opposite side = sin(x) * hypotenuse = 0.6 * 10 = 6 cm

Question 2

A projectile is launched at an angle of 30 degrees to the horizontal. Find the range of the projectile using the tangent function.

Solution: Using the tangent function, we can calculate the range of the projectile as follows:

tan(x) = opposite side / adjacent side range = (tan(x) * initial velocity)^2 / (2 * acceleration due to gravity) = (tan(30) * 10)^2 / (2 * 9.8) = 25 m

Question 3

A bridge has a length of 100 m and a height of 20 m. Find the angle of the bridge using the cosine function.

Solution: Using the cosine function, we can calculate the angle of the bridge as follows:

cos(x) = adjacent side / hypotenuse angle = arccos(cos(x)) = arccos(0.8) = 36.87 degrees

NCERT Textbook Questions & Detailed Answers

Question 1

A right-angled triangle has a hypotenuse of length 10 cm and an adjacent side of length 6 cm. Find the length of the opposite side using the sine function.

Solution: Using the sine function, we can calculate the length of the opposite side as follows:

sin(x) = opposite side / hypotenuse opposite side = sin(x) * hypotenuse = 0.6 *