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 triangle to the ratios of its sides. In Class 11 NCERT Mathematics, these functions form the foundation for understanding periodic phenomena and solving complex problems in geometry and calculus.

The six primary trigonometric functions are:

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

These functions are defined using a right-angled triangle where one angle is $\theta$. For example, $\sin \theta$ is the ratio of the length of the side opposite to $\theta$ to the hypotenuse.

Definitions and Ratios of Trigonometric Functions

In a right-angled triangle ABC, with $\angle B = 90^{\circ}$ and $\theta = \angle A$, the trigonometric functions are defined as:

These ratios help in solving various problems involving triangles and angles.

Important Formulas and Identities in Trigonometric Functions

Class 11 NCERT Mathematics introduces key trigonometric identities that simplify problem-solving:

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:

$$\sin \theta = \frac{1}{\csc \theta}$$ $$\cos \theta = \frac{1}{\sec \theta}$$ $$\tan \theta = \frac{1}{\cot \theta}$$

3. Quotient Identities:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$

These formulas are essential for simplifying expressions and solving equations in exams.

Worked Example: Finding Trigonometric Ratios

Example: In a right-angled triangle, if the side opposite to angle $\theta$ is 3 cm and the hypotenuse is 5 cm, find $\sin \theta$, $\cos \theta$, and $\tan \theta$.

Solution:

• $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5} = 0.6$
• Using Pythagoras theorem, adjacent side = $\sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ cm
• $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} = 0.8$
• $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4} = 0.75$

This example shows how to calculate basic trigonometric ratios from triangle sides.

Graphs and Properties of Trigonometric Functions

Understanding the graphs of trigonometric functions helps visualize their behaviour:

• Sine and Cosine graphs: Periodic with period $2\pi$, oscillate between -1 and 1.
• Tangent graph: Periodic with period $\pi$, has vertical asymptotes where function is undefined.

These properties are crucial for solving trigonometric equations and understanding periodic phenomena.

Applications of Trigonometric Functions in Class 11

Trigonometric functions are widely used in Class 11 Mathematics and beyond:

• Solving problems involving heights and distances
• Calculating angles and sides in triangles
• Understanding wave motion and oscillations
• Basis for calculus topics like derivatives and integrals of trig functions

Mastering these functions helps build a strong foundation for higher studies in science, engineering, and technology.