What is Trigonometric Functions Class 11: Definition & Basics

Introduction to Trigonometric Functions in Class 11

Trigonometric functions are mathematical functions that relate the angles of a right-angled triangle to the ratios of its sides. In Class 11 NCERT Mathematics, these functions form the foundation for studying periodic phenomena and advanced geometry.

The six basic trigonometric functions are:

• Sine ($\sin$)
• Cosine ($\cos$)
• Tangent ($\tan$)
• Cosecant ($\csc$)
• Secant ($\sec$)
• Cotangent ($\cot$)

Each function is defined as a ratio of two sides of a right-angled triangle relative to one of its acute angles. Understanding these ratios helps solve problems involving angles and distances.

Definitions of Six Trigonometric Functions

In a right-angled triangle, consider an acute angle $\theta$. The sides are named as follows:

• Hypotenuse: the side opposite the right angle (longest side)
• Opposite side: the side opposite to angle $\theta$
• Adjacent side: the side next to angle $\theta$ (other than hypotenuse)

The six trigonometric functions are defined as:

These definitions are the building blocks for solving trigonometric problems.

Understanding the Unit Circle and Angle Measurement

While the right-angled triangle definition is fundamental, Class 11 NCERT also introduces the unit circle approach to trigonometric functions.

• The unit circle is a circle with radius 1 centered at the origin of the coordinate plane.
• An angle $\theta$ is measured from the positive x-axis.
• The coordinates of a point on the unit circle corresponding to angle $\theta$ are $(\cos \theta, \sin \theta)$.

This approach helps extend trigonometric functions beyond acute angles (0° to 90°) to all real numbers.

Angle Measurement:

• Angles can be measured in degrees or radians.
• $180^\circ = \pi$ radians.

Understanding this concept is essential for advanced topics like trigonometric graphs and identities.

Important Trigonometric Identities for Class 11 Students

Trigonometric identities are equations involving trigonometric functions that hold true for all valid values of the variables. Key identities include:

1. Pythagorean Identities: $$\sin^2 \theta + \cos^2 \theta = 1$$ $$1 + \tan^2 \theta = \sec^2 \theta$$ $$1 + \cot^2 \theta = \csc^2 \theta$$

2. Reciprocal Identities: $$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}$$

3. Quotient Identities: $$\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$$

These identities simplify solving equations and proving other mathematical results.

Worked Example: Finding Trigonometric Ratios

Example: In a right-angled triangle, if the side opposite to angle $\theta$ is 3 cm and the adjacent side is 4 cm, find all six trigonometric functions of $\theta$.

Solution:

1. Calculate the hypotenuse using Pythagoras theorem: $$\text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$$

2. Calculate the functions:

• $\sin \theta = \frac{3}{5} = 0.6$
• $\cos \theta = \frac{4}{5} = 0.8$
• $\tan \theta = \frac{3}{4} = 0.75$
• $\csc \theta = \frac{5}{3} \approx 1.67$
• $\sec \theta = \frac{5}{4} = 1.25$
• $\cot \theta = \frac{4}{3} \approx 1.33$

This example illustrates how to use side lengths to find trigonometric ratios.

Applications of Trigonometric Functions in Class 11

Trigonometric functions are widely used in various fields, and Class 11 NCERT highlights their importance in:

• Solving problems involving heights and distances
• Analyzing periodic phenomena like sound and light waves
• Engineering and physics for resolving forces
• Geometry problems involving angles and circles

Mastering these functions in Class 11 lays the foundation for calculus, physics, and engineering courses.