Class 11 Trigonometric Functions – Core Concepts for JEE, NEET & KCET/COMEDK

This chapter builds on the foundation of angles and introduces Trigonometric Functions, which are central to solving complex problems in calculus, physics, and engineering. It links directly to the Unit: Trigonometric Functions.

Competitive Exam Marks Distribution

Exam	Approximate Weightage
JEE Main	2–3 questions (~8–12 marks)
NEET (Physics)	2–3 marks (angular functions, wave motion)
KCET/COMEDK	1–2 direct questions

What are Trigonometric Functions?

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. These are essential tools in mathematics and physics, especially when dealing with rotational systems, waves, and periodic phenomena. The six primary trigonometric functions are:

Sine ( $\sin$ )
Cosine ( $\cos$ )
Tangent ( $\tan$ )
Cosecant ( $\csc$ or $\text{cosec}$ )
Secant ( $\sec$ )
Cotangent ( $\cot$ )

These functions can also be defined using the unit circle and are periodic in nature.

Definitions in a Right Triangle:

Let the triangle have angle $\theta$ , then:

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$
$\csc \theta = \frac{1}{\sin \theta}$
$\sec \theta = \frac{1}{\cos \theta}$
$\cot \theta = \frac{1}{\tan \theta}$

These ratios do not depend on the size of the triangle, making them universal for any triangle with a given angle.

Trigonometric Functions on the Unit Circle

In the unit circle (radius = 1):

The coordinates of a point at angle $\theta$ are $(\cos \theta, \sin \theta)$ .
Trigonometric functions become defined for all real values of $\theta$ .
The values of these functions repeat every $2\pi$ , which illustrates their periodic nature.

Example:
$\cos(\pi) = -1$ , $\sin(\pi) = 0$ , $\tan(\pi) = 0$

This circular interpretation extends the domain of trigonometric functions from acute angles to all real numbers.

Sign of Trigonometric Functions (ASTC Rule)

Understanding which trigonometric functions are positive in which quadrants is essential for evaluating expressions and solving equations. This is summarized using the mnemonic “All Students Take Calculus.”

Quadrant	Positive Functions	Mnemonic
I	All functions	All
II	$\sin$ , $\csc$	Students
III	$\tan$ , $\cot$	Take
IV	$\cos$ , $\sec$	Calculus

Table – Signs of Trigonometric Functions in Each Quadrant

	I	II	III	IV
$\sin x$	+	+	–	–
$\cos x$	+	–	–	+
$\tan x$	+	–	+	–
$\csc x$	+	+	–	–
$\sec x$	+	–	–	+
$\cot x$	+	–	+	–

Periodicity and Domain/Range

Trigonometric functions are periodic. Their graphs repeat over specific intervals known as periods. Here’s a summary:

Function	Period	Domain	Range
$\sin x$	$2\pi$	$\mathbb{R}$	$[-1, 1]$
$\cos x$	$2\pi$	$\mathbb{R}$	$[-1, 1]$
$\tan x$	$\pi$	$\mathbb{R} \setminus {(2n+1)\frac{\pi}{2}}$	$\mathbb{R}$

Table – Behaviour of Trigonometric Functions in Each Quadrant

Function	I quadrant	II quadrant	III quadrant	IV quadrant
$\sin$	increases from 0 to 1	decreases from 1 to 0	decreases from 0 to –1	increases from –1 to 0
$\cos$	decreases from 1 to 0	decreases from 0 to –1	increases from –1 to 0	increases from 0 to 1
$\tan$	increases from 0 to $\infty$	increases from $-\infty$ to 0	increases from 0 to $\infty$	increases from $-\infty$ to 0
$\cot$	decreases from $\infty$ to 0	decreases from 0 to $-\infty$	decreases from $\infty$ to 0	decreases from 0 to $-\infty$
$\sec$	increases from 1 to $\infty$	increases from $-\infty$ to –1	increases from $-\infty$ to –1	decreases from $\infty$ to 1
$\csc$	decreases from $\infty$ to 1	increases from 1 to $\infty$	increases from $-\infty$ to –1	decreases from $-\infty$ to –1

Trigonometric Identities

These identities are essential in simplifying expressions, proving equations, and solving integrals.

Fundamental Identity:

$\sin^2 \theta + \cos^2 \theta = 1$

Derived Identities:

$1 + \tan^2 \theta = \sec^2 \theta$
$1 + \cot^2 \theta = \csc^2 \theta$

These formulas form the backbone of most trigonometric transformations.

Example Problems

Q1. Find $\sin(\frac{\pi}{3})$ and $\cos(\frac{\pi}{3})$ .

$\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
$\cos(\frac{\pi}{3}) = \frac{1}{2}$

Q2. Prove $\tan \theta = \frac{\sin \theta}{\cos \theta}$ .

A: From definitions: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin \theta}{\cos \theta}$

Practice Questions

Find all values of $x \in [0, 2\pi]$ such that $\sin x = \frac{1}{2}$ .
Evaluate $\tan(\frac{3\pi}{4})$ .
Prove $1 + \tan^2 x = \sec^2 x$ using the identity.
Sketch the graph of $\sin x$ over $[0, 2\pi]$ .

FAQs

Q1. Why are trigonometric functions important in physics?
They describe oscillations, waves, rotational motion, and electric circuits.

Q2. How can I remember trigonometric identities easily?
Use the unit circle and mnemonic devices like “All Students Take Calculus.”

Q3. Which function is undefined at $\frac{\pi}{2}$ ?
$\tan(\frac{\pi}{2})$ and $\sec(\frac{\pi}{2})$ are undefined.

Q4. How do I memorize signs in each quadrant?
Remember ASTC — All, Sine, Tangent, Cosine.

Q5. What is the difference between $\boldsymbol{(x, y)}$ and trigonometric coordinates?
In the unit circle, a point $(x, y)$ represents $(\cos \theta, \sin \theta)$ .

Conclusion

Trigonometric functions are at the core of mathematical applications in science, engineering, and technology. From waves and circular motion in NEET to calculus and graph analysis in JEE and KCET/COMEDK, these functions are indispensable. A strong foundation in identities, properties, and graphical behavior will significantly boost your problem-solving skills in competitive exams. Practice identities, understand the unit circle, and solve diverse problems to

3.3 Trigonometric Functions – Class 11 Mathematics

Competitive Exam Marks Distribution

What are Trigonometric Functions?

Definitions in a Right Triangle:

Trigonometric Functions on the Unit Circle

Sign of Trigonometric Functions (ASTC Rule)

Table – Signs of Trigonometric Functions in Each Quadrant

Periodicity and Domain/Range

Table – Behaviour of Trigonometric Functions in Each Quadrant

Trigonometric Identities

Fundamental Identity:

Derived Identities:

Example Problems

Practice Questions

FAQs

Q1. Why are trigonometric functions important in physics? They describe oscillations, waves, rotational motion, and electric circuits.

Q2. How can I remember trigonometric identities easily? Use the unit circle and mnemonic devices like “All Students Take Calculus.”

Q3. Which function is undefined at ? and are undefined.

Q4. How do I memorize signs in each quadrant? Remember ASTC — All, Sine, Tangent, Cosine.

Q5. What is the difference between and trigonometric coordinates? In the unit circle, a point represents .

Conclusion

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Q1. Why are trigonometric functions important in physics?
They describe oscillations, waves, rotational motion, and electric circuits.

Q2. How can I remember trigonometric identities easily?
Use the unit circle and mnemonic devices like “All Students Take Calculus.”

Q3. Which function is undefined at $\frac{\pi}{2}$ ?
$\tan(\frac{\pi}{2})$ and $\sec(\frac{\pi}{2})$ are undefined.

Q4. How do I memorize signs in each quadrant?
Remember ASTC — All, Sine, Tangent, Cosine.

Q5. What is the difference between $\boldsymbol{(x, y)}$ and trigonometric coordinates?
In the unit circle, a point $(x, y)$ represents $(\cos \theta, \sin \theta)$ .