Trigonometric Functions Class 11 Notes (2026–27) | Easy & Complete Guide with Identities

🔰 Introduction to Trigonometric Functions

Trigonometric Functions Class 11 is one of the most important chapters in mathematics that introduces the relationship between angles and sides of a triangle. Trigonometry mainly deals with six fundamental functions—sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). These functions help us understand how angles influence distances and positions in both theoretical and real-world situations.

In real life, trigonometry is widely used in calculating heights and distances, such as finding the height of a building or the distance of a ship from the shore. It also plays a crucial role in physics, especially in wave motion, sound, light, and oscillations. Engineers, architects, and even astronomers rely on trigonometric concepts for precise calculations.

For students preparing for CBSE board exams, JEE, or other competitive exams, Trigonometric Functions Class 11 forms the foundation for advanced topics like calculus, vectors, and coordinate geometry. A strong understanding of this chapter not only improves problem-solving skills but also boosts overall mathematical confidence.

trigonometric functions class 11 is one of the most important chapters in NCERT Mathematics, covering formulas, graphs, identities, and problem-solving techniques.

 📐 2. Angles and Their Measurement

🔹 Degree Measure vs Radian Measure

In Trigonometric Functions Class 11, angles can be measured in two systems:

• Degree Measure (°): A full circle is divided into 360 equal parts, so one complete revolution = 360°.
• Radian Measure (rad): A full circle is equal to 2π radians, based on the radius of the circle.

Key Relation:

π radians = 180°

Example 2: Convert π/2 radians into degrees

π/2 = (180 / π) × (π / 2) = 90°

📌 Concept Clarity

If rotation starts from the positive x-axis:

• Anticlockwise → Positive angle
• Clockwise → Negative angle

Example:

• +180° → Half turn anticlockwise
• -180° → Half turn clockwise

Both reach the same final position but follow different directions.

In trigonometric functions class 11, understanding the unit circle helps you visualize values of sin, cos, and tan.

The graphs in trigonometric functions class 11 are essential for exams and concept clarity.

🔄 3. Trigonometric Functions (Basic Definitions)

In Trigonometric Functions Class 11, we study six basic trigonometric functions that relate the angles of a triangle to the ratios of its sides. These functions are fundamental for solving problems in mathematics, physics, and engineering.

📐 Six Trigonometric Functions

• sin θ (Sine)
• cos θ (Cosine)
• tan θ (Tangent)
• cosec θ (Cosecant)
• sec θ (Secant)
• cot θ (Cotangent)

• sin θ = Perpendicular / Hypotenuse
• cos θ = Base / Hypotenuse
• tan θ = Perpendicular / Base
• cosec θ = Hypotenuse / Perpendicular
• sec θ = Hypotenuse / Base
• cot θ = Base / Perpendicular

⭕ Unit Circle Introduction

A unit circle is a circle with:

• Radius = 1
• Center at O(0, 0)

📌 Key Concept

If a point P lies on the unit circle making an angle θ with the positive x-axis, then:

P = (cos θ, sin θ)

This means:

• cos θ = x-coordinate
• sin θ = y-coordinate

📌 Why Unit Circle is Important?

• Defines trigonometric functions for all angles
• Works for positive and negative angles
• Helps in understanding graphs and identities

Right triangle definitions are useful for basic understanding, but the unit circle concept is essential for advanced topics like trigonometric equations and graphs in Trigonometric Functions Class 11.

🔵 4. Trigonometric Functions on Unit Circle

In Trigonometric Functions Class 11, the unit circle helps us find the values of trigonometric functions for standard angles.
Every point on the unit circle is represented as:

(cos θ, sin θ)

This means:

• cos θ = x-coordinate
• sin θ = y-coordinate

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📍 Coordinates of Standard Angles

Angle	Coordinates (cos θ, sin θ)	sin θ	cos θ	tan θ
0°	(1, 0)	0	1	0
30°	(√3/2, 1/2)	1/2	√3/2	1/√3
45°	(1/√2, 1/√2)	1/√2	1/√2	1
60°	(1/2, √3/2)	√3/2	1/2	√3
90°	(0, 1)	1	0	Not Defined

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📌 Important Observations

• As angle increases, sin θ increases and cos θ decreases
• tan θ = sin θ / cos θ
• tan 90° is not defined because cos 90° = 0

⭐ These values are extremely important for board exams and competitive exams like JEE.

5. Graphs of Trigonometric Functions

🔵 Graph of sin x

Note- Students are advised to use the Above ( Last ) graph for exam which is the easiest one. However , other graphs can be used for more concept.

✅ Key Points:

• sin 0 = 0
• sin (π/2) = 1
• sin π = 0
• sin (3π/2) = -1
• sin 2π = 0

📌 Important Features:

• Range: [-1, 1]
• Period: 2π
• Wave starts from origin

🔴 Graph of cos x

 

🟢 Graph of tan x

🟢 Key Points (tan x)

• tan 0 = 0
• tan (π/4) = 1
• tan (-π/4) = -1

📌 Important Features

• Range: (-∞, +∞)
• Period: π
• Undefined at:
  • x = π/2
  • x = 3π/2
  • … (odd multiples of π/2)

👉 These points are called vertical asymptotes, where the graph of tan x is not defined.

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🔁 6. Trigonometric Identities (Class 11)

In Trigonometric Functions Class 11, identities are equations that are true for all values of θ (where defined).

📌 1. Reciprocal Identities

• sin θ = 1 / cosec θ
• cos θ = 1 / sec θ
• tan θ = 1 / cot θ
• cosec θ = 1 / sin θ
• sec θ = 1 / cos θ
• cot θ = 1 / tan θ

📌 2. Quotient Identities

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

📌 3. Pythagorean Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

📌 4. Trigonometric Identities (Derived Forms)

• 1 − sin²θ = cos²θ
• 1 − cos²θ = sin²θ
• sec²θ − 1 = tan²θ
• cosec²θ − 1 = cot²θ

📌 5. Allied Angle Identities

• sin (90° − θ) = cos θ
• cos (90° − θ) = sin θ
• tan (90° − θ) = cot θ
• cot (90° − θ) = tan θ
• sec (90° − θ) = cosec θ
• cosec (90° − θ) = sec θ

📌 6. Negative Angle Identities

• sin (−θ) = −sin θ
• cos (−θ) = cos θ
• tan (−θ) = −tan θ
• cosec (−θ) = −cosec θ
• sec (−θ) = sec θ
• cot (−θ) = −cot θ

📌 7. Periodicity Identities

📌 Trigonometric Identities (All Quadrants)

These identities help determine the values of trigonometric functions in different quadrants using reference angles.

🔄 Quadrant-wise Identities Table

Quadrant	Angle Form	sin	cos	tan
1st Quadrant	θ	sin θ	cos θ	tan θ
2nd Quadrant	(π − θ)	sin θ	−cos θ	−tan θ
3rd Quadrant	(π + θ)	−sin θ	−cos θ	tan θ
4th Quadrant	(2π − θ)	−sin θ	cos θ	−tan θ

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📌 Complete Identities (All Functions)

Angle	sin	cos	tan	cosec	sec	cot
π − θ	sin θ	−cos θ	−tan θ	cosec θ	−sec θ	−cot θ
π + θ	−sin θ	−cos θ	tan θ	−cosec θ	−sec θ	cot θ
2π − θ	−sin θ	cos θ	−tan θ	−cosec θ	sec θ	−cot θ

📌 Quadrant Formulae (90° ± θ, 270° ± θ)

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🔴 (90° + θ)

Function	Result
sin (90° + θ)	cos θ
cos (90° + θ)	−sin θ
tan (90° + θ)	−cot θ
cot (90° + θ)	−tan θ
sec (90° + θ)	−cosec θ
cosec (90° + θ)	sec θ

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🔵 (270° − θ)

Function	Result
sin (270° − θ)	−cos θ
cos (270° − θ)	−sin θ
tan (270° − θ)	cot θ
cot (270° − θ)	tan θ
sec (270° − θ)	−cosec θ
cosec (270° − θ)	−sec θ

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🟢 (270° + θ)

Function	Result
sin (270° + θ)	−cos θ
cos (270° + θ)	sin θ
tan (270° + θ)	−cot θ
cot (270° + θ)	−tan θ
sec (270° + θ)	cosec θ
cosec (270° + θ)	−sec θ

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👉 Shortcut Rule:
90° ± θ → sin ↔ cos (co-function change)
270° ± θ → use reference angle + sign from quadrant

📌 Quadrant Formulae (π Form)

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🔴 (π/2 + θ)

Function	Result
sin (π/2 + θ)	cos θ
cos (π/2 + θ)	−sin θ
tan (π/2 + θ)	−cot θ
cot (π/2 + θ)	−tan θ
sec (π/2 + θ)	−cosec θ
cosec (π/2 + θ)	sec θ

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🔵 (3π/2 − θ)

Function	Result
sin (3π/2 − θ)	−cos θ
cos (3π/2 − θ)	−sin θ
tan (3π/2 − θ)	cot θ
cot (3π/2 − θ)	tan θ
sec (3π/2 − θ)	−cosec θ
cosec (3π/2 − θ)	−sec θ

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🟢 (3π/2 + θ)

Function	Result
sin (3π/2 + θ)	−cos θ
cos (3π/2 + θ)	sin θ
tan (3π/2 + θ)	−cot θ
cot (3π/2 + θ)	−tan θ
sec (3π/2 + θ)	cosec θ
cosec (3π/2 + θ)	−sec θ

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👉 Shortcut:
π/2 ± θ → sin ↔ cos (co-function change)
3π/2 ± θ → use reference angle + ASTC rule 

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👉 Shortcut Trick (ASTC Rule):
A → All positive (1st quadrant)
S → Sine positive (2nd quadrant)
T → Tangent positive (3rd quadrant)
C → Cosine positive (4th quadrant)

Note-

Sign Rule (positive / negative) holds good for the reciprocal trigonometric functions also.

➕ Sum of Angles Formulae

In Trigonometric Functions Class 11, sum of angles identities are used to find values of trigonometric functions when angles are added.

📌 Basic Sum Formulae

• sin (A + B) = sin A cos B + cos A sin B
• cos (A + B) = cos A cos B − sin A sin B
• tan (A + B) = (tan A + tan B) / (1 − tan A tan B)
• cot (A + B) = (cot A cot B − 1) / (cot A + cot B)

📌 Important Observations

• sin uses + sign in between
• cos uses − sign in between
• tan formula is in fraction form

📌 Special Cases

Expression	Result
sin (A + 90°)	cos A
cos (A + 90°)	−sin A
tan (A + 90°)	−cot A

📌 Quick Trick

👉 Remember:
sin → +
cos → −
tan → fraction formula

➖ Difference of Angles Formulae

In Trigonometric Functions Class 11, difference of angles identities are used to evaluate trigonometric functions when angles are subtracted.

📌 Basic Difference Formulae

• sin (A − B) = sin A cos B − cos A sin B
• cos (A − B) = cos A cos B + sin A sin B
• tan (A − B) = (tan A − tan B) / (1 + tan A tan B)
• cot (A − B) = (cot A cot B + 1) / (cot B − cot A)

📌 Important Observations

• sin uses − sign in between
• cos uses + sign in between
• tan and cot are in fraction form

📌 Special Cases

Expression	Result
sin (A − 90°)	−cos A
cos (A − 90°)	sin A
tan (A − 90°)	−cot A
cot (A − 90°)	−tan A

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👉 Quick Trick:
sin → minus
cos → plus
tan → (− / +)
cot → (+ / −)

➕ Sum of Trigonometric Ratios (T.R.s)

These identities help to convert the sum of sine and cosine into a single trigonometric function.

🔁 Sum to Product Formulae (sin A + sin B Type)

These identities convert the sum or difference of trigonometric functions into a product form.

📌 Sine Formulas

• sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)
• sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)

📌 Cosine Formulas

• cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)
• cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

📌 Quick Pattern (Very Important)

👉 sin + sin → sin × cos
👉 sin − sin → cos × sin
👉 cos + cos → cos × cos
👉 cos − cos → − sin × sin

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📌 Example

sin 70° + sin 50°

= 2 sin( (70 + 50)/2 ) cos( (70 − 50)/2 )
= 2 sin(60°) cos(10°)

⭐ These formulas are very important for simplifying expressions and solving equations in board exams and JEE.

📌 Standard Forms

• a sin x + b cos x = R sin (x + α)
• a cos x + b sin x = R cos (x − α)

where:

• R = √(a² + b²)
• tan α = b / a

📌 Special Case

• sin x + cos x = √2 sin (x + π/4)
• sin x + cos x = √2 cos (x − π/4)

📌 Important Observations

• Any expression of form a sin x + b cos x can be written as a single sine or cosine
• This is useful in maximum/minimum value problems

👉 Shortcut:
R = √(a² + b²) and tan α = b/a

🔁 Double Angle Formulae

In Trigonometric Functions Class 11, double angle formulas are used to express trigonometric functions of 2A in terms of A.

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🔴 sin 2A

• sin 2A = 2 sin A cos A
• sin 2A = (2 tan A) / (1 + tan²A)

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🔵 cos 2A

• cos 2A = cos²A − sin²A
• cos 2A = 2cos²A − 1
• cos 2A = 1 − 2sin²A
• cos 2A = (1 − tan²A) / (1 + tan²A)

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🟢 tan 2A

• tan 2A = (2 tan A) / (1 − tan²A)

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🟣 cot 2A

• cot 2A = (cot²A − 1) / (2 cot A)

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👉 Quick Pattern:
sin 2A → depends on sin & cos OR tan
cos 2A → multiple forms (MOST IMPORTANT)
tan 2A → fraction formula (very common in exams)

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📌 Special Values

Angle (A)	sin 2A	cos 2A	tan 2A
30°	sin 60° = √3/2	cos 60° = 1/2	tan 60° = √3
45°	sin 90° = 1	cos 90° = 0	tan 90° = Not Defined

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👉 Quick Trick:
sin 2A → 2 sin A cos A
cos 2A → 3 forms (most important)
tan 2A → fraction formula

📌 Half Angle Formulae

🔴 Forms of sin A (Using Half Angle)

Formula
sin A = 2 sin(A/2) cos(A/2)
sin A = (2 tan(A/2)) / (1 + tan²(A/2))

🔵 Forms of cos A (Using Half Angle)

Formula
cos A = 2cos²(A/2) − 1
cos A = 1 − 2sin²(A/2)
cos A = (1 − tan²(A/2)) / (1 + tan²(A/2))

🟢 Forms of tan A (Using Half Angle)

Formula
tan A = (2 tan(A/2)) / (1 − tan²(A/2))
tan A = (2 sin(A/2) cos(A/2)) / (cos²(A/2) − sin²(A/2))

🔴 sin 3A

Formula
sin 3A = 3 sin A − 4 sin³A
sin 3A = 4 sin A cos²A − sin A

🔵 cos 3A

Formula
cos 3A = 4cos³A − 3cos A
cos 3A = cos A (4cos²A − 3)

🟢 tan 3A

Formula
tan 3A = (3tan A − tan³A) / (1 − 3tan²A)

🔥 Quick Trick (Important)

• sin 3A → 3s − 4s³
• cos 3A → 4c³ − 3c
• tan 3A → (3t − t³) / (1 − 3t²)

In conclusion, mastering trigonometric functions class 11 is essential for building a strong foundation in mathematics.

For authentic theory and examples, refer to the
NCERT Class 11 Maths Chapter – Trigonometric Functions

You can check the latest syllabus from the
CBSE Official Curriculum Website