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Absolutely. Here’s a Class 11 NCERT-style Chapter 3: Trigonometric Functions summary, arranged so you can use it both for exam revision and as a foundation for Class 12/physics.

📘 Class 11 Maths — Chapter 3: Trigonometric Functions

Core idea: Trigonometry is about relating angles to ratios of sides, and then extending those relationships to all angles using the unit circle.

1. Angles and Their Measurement

An angle measures the amount of rotation of a ray around a fixed point.

Two common unitsDegrees

A complete rotation:

360∘360^\circ

Therefore,

180∘=straight angle180^\circ = \text{straight angle}90∘=right angle90^\circ = \text{right angle}Radians ⭐ IMPORTANT

A radian is defined using a circle.

If the length of an arc is equal to the radius of the circle, the angle made at the centre is 1 radian.

For a complete circle:

2π radians=360∘2\pi\text{ radians}=360^\circ

Therefore:

180∘=π radians\boxed{180^\circ=\pi\text{ radians}}Conversion formulas ⭐ IMPORTANT

Degrees → radians:

θ∘×π180\boxed{\theta^\circ\times\frac{\pi}{180}}

Radians → degrees:

θ×180π\boxed{\theta\times\frac{180}{\pi}}

Examples:

60∘=60×π180=π360^\circ=60\times\frac{\pi}{180} =\boxed{\frac{\pi}{3}}

and

π4=π4×180π=45∘\frac{\pi}{4}=\frac{\pi}{4}\times\frac{180}{\pi} =\boxed{45^\circ}

2. Relation Between Arc Length and Angle ⭐

For a circle of radius rr, if an angle is θ\theta radians:

l=rθ\boxed{l=r\theta}

where:

• ll = arc length

• rr = radius

• θ\theta = angle in radians

VERY IMPORTANT

The formula l=rθl=r\theta requires θ\theta to be in radians.

Example:

If r=7r=7 cm and θ=π3\theta=\frac{\pi}{3},

l=7(π3)l=7\left(\frac{\pi}{3}\right)l=7π3 cm\boxed{l=\frac{7\pi}{3}\text{ cm}}

3. Types of AnglesPositive angle

Anticlockwise rotation.

Negative angle

Clockwise rotation.

For example:

−60∘-60^\circ

means rotate 60∘60^\circ clockwise.

Quadrants ⭐ IMPORTANT

The coordinate plane is divided into four quadrants:

Or in radians:

4. Trigonometric Functions

For an acute angle θ\theta in a right triangle:

sin⁡θ=PerpendicularHypotenuse\boxed{\sin\theta=\frac{\text{Perpendicular}}{\text{Hypotenuse}}}cos⁡θ=BaseHypotenuse\boxed{\cos\theta=\frac{\text{Base}}{\text{Hypotenuse}}}tan⁡θ=PerpendicularBase\boxed{\tan\theta=\frac{\text{Perpendicular}}{\text{Base}}}

The reciprocal functions:

cosec⁡θ=1sin⁡θ\boxed{\cosec\theta=\frac{1}{\sin\theta}}sec⁡θ=1cos⁡θ\boxed{\sec\theta=\frac{1}{\cos\theta}}cot⁡θ=1tan⁡θ\boxed{\cot\theta=\frac{1}{\tan\theta}}

Also:

tan⁡θ=sin⁡θcos⁡θ\boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}}cot⁡θ=cos⁡θsin⁡θ\boxed{\cot\theta=\frac{\cos\theta}{\sin\theta}}⭐ IMPORTANT

Memorise this relationship:

sin⁡θ,cos⁡θ,tan⁡θ\boxed{\sin\theta,\cos\theta,\tan\theta}

are the main three.

The other three are their reciprocals.

5. Trigonometric Functions for Any Angle ⭐⭐⭐

This is where Class 11 becomes different from basic triangle trigonometry.

We use the coordinate plane/unit circle to define trig functions for angles like:

120∘,225∘,−60∘,450∘120^\circ,\quad 225^\circ,\quad -60^\circ,\quad 450^\circ

So trigonometric functions are not restricted to 0∘0^\circ–90∘90^\circ.

6. Signs of Trigonometric Functions ⭐⭐⭐

This is very important for exams.

Easy memory:

All Students Take Calculus

• All → Quadrant I

• Students → sin positive

• Take → tan positive

• Calculus → cos positive

Meaning:

QI: all positive
QII: only sin positive
QIII: only tan positive
QIV: only cos positive

Since reciprocal functions have the same sign as their corresponding functions:

• sin ↔ cosec

• cos ↔ sec

• tan ↔ cot

7. Standard Values ⭐⭐⭐

You absolutely need these.

⭐ Exam importance: VERY HIGH

Don't just memorise the table blindly.

Notice:

sin⁡θ=0,12,12,32,1\sin\theta = 0,\frac12,\frac1{\sqrt2},\frac{\sqrt3}{2},1

as the angle increases from 0∘0^\circ to 90∘90^\circ.

And cosine goes in reverse:

1,32,12,12,01,\frac{\sqrt3}{2},\frac1{\sqrt2},\frac12,0

8. Fundamental Trigonometric Identities ⭐⭐⭐

These are some of the most important formulas in the entire chapter.

Identity 1sin⁡2θ+cos⁡2θ=1\boxed{\sin^2\theta+\cos^2\theta=1}

From this:

sin⁡2θ=1−cos⁡2θ\boxed{\sin^2\theta=1-\cos^2\theta}cos⁡2θ=1−sin⁡2θ\boxed{\cos^2\theta=1-\sin^2\theta}

Identity 2

Divide the first identity by cos⁡2θ\cos^2\theta:

1+tan⁡2θ=sec⁡2θ\boxed{1+\tan^2\theta=\sec^2\theta}

Therefore:

tan⁡2θ=sec⁡2θ−1\boxed{\tan^2\theta=\sec^2\theta-1}

Identity 3

Divide the first identity by sin⁡2θ\sin^2\theta:

1+cot⁡2θ=cosec⁡2θ\boxed{1+\cot^2\theta=\cosec^2\theta}

Therefore:

cot⁡2θ=cosec⁡2θ−1\boxed{\cot^2\theta=\cosec^2\theta-1}⭐⭐⭐ EXAM ALERT

These three should become automatic:

sin⁡2θ+cos⁡2θ=1\boxed{\sin^2\theta+\cos^2\theta=1}1+tan⁡2θ=sec⁡2θ\boxed{1+\tan^2\theta=\sec^2\theta}1+cot⁡2θ=cosec⁡2θ\boxed{1+\cot^2\theta=\cosec^2\theta}

A huge number of simplification/proof questions are built around them.

9. Even and Odd Functions ⭐⭐Sinesin⁡(−θ)=−sin⁡θ\boxed{\sin(-\theta)=-\sin\theta}

So sine is an odd function.

Cosinecos⁡(−θ)=cos⁡θ\boxed{\cos(-\theta)=\cos\theta}

Cosine is an even function.

Tangenttan⁡(−θ)=−tan⁡θ\boxed{\tan(-\theta)=-\tan\theta}

Tangent is odd.

Similarly:

cosec⁡(−θ)=−cosec⁡θ\boxed{\cosec(-\theta)=-\cosec\theta}sec⁡(−θ)=sec⁡θ\boxed{\sec(-\theta)=\sec\theta}cot⁡(−θ)=−cot⁡θ\boxed{\cot(-\theta)=-\cot\theta}Quick memory

sin, tan, cosec, cot → odd

cos, sec → even

10. Periodicity ⭐⭐⭐

A periodic function repeats its values.

For sine:

sin⁡(θ+2nπ)=sin⁡θ\boxed{\sin(\theta+2n\pi)=\sin\theta}

For cosine:

cos⁡(θ+2nπ)=cos⁡θ\boxed{\cos(\theta+2n\pi)=\cos\theta}

For tangent:

tan⁡(θ+nπ)=tan⁡θ\boxed{\tan(\theta+n\pi)=\tan\theta}

where n∈Zn\in\mathbb Z.

PeriodsPeriod of sin⁡θ=2π\boxed{\text{Period of }\sin\theta=2\pi}Period of cos⁡θ=2π\boxed{\text{Period of }\cos\theta=2\pi}Period of tan⁡θ=π\boxed{\text{Period of }\tan\theta=\pi}

Likewise:

Period of cosec⁡θ=2π\boxed{\text{Period of }\cosec\theta=2\pi}Period of sec⁡θ=2π\boxed{\text{Period of }\sec\theta=2\pi}Period of cot⁡θ=π\boxed{\text{Period of }\cot\theta=\pi}⭐ Future importance

This becomes extremely useful when you study graphs, equations and calculus.

11. Allied Angles ⭐⭐⭐

These are angles related to:

90∘,180∘,270∘,360∘90^\circ,\quad180^\circ,\quad270^\circ,\quad360^\circ

or

π2,π,3π2,2π\frac{\pi}{2},\quad\pi,\quad\frac{3\pi}{2},\quad2\pi

The main idea:

Convert the complicated angle into a familiar reference angle and determine the sign from its quadrant.

Important formulas90∘−θ90^\circ-\thetasin⁡(90∘−θ)=cos⁡θ\boxed{\sin(90^\circ-\theta)=\cos\theta}cos⁡(90∘−θ)=sin⁡θ\boxed{\cos(90^\circ-\theta)=\sin\theta}tan⁡(90∘−θ)=cot⁡θ\boxed{\tan(90^\circ-\theta)=\cot\theta}

90∘+θ90^\circ+\thetasin⁡(90∘+θ)=cos⁡θ\boxed{\sin(90^\circ+\theta)=\cos\theta}cos⁡(90∘+θ)=−sin⁡θ\boxed{\cos(90^\circ+\theta)=-\sin\theta}tan⁡(90∘+θ)=−cot⁡θ\boxed{\tan(90^\circ+\theta)=-\cot\theta}

180∘−θ180^\circ-\thetasin⁡(180∘−θ)=sin⁡θ\boxed{\sin(180^\circ-\theta)=\sin\theta}cos⁡(180∘−θ)=−cos⁡θ\boxed{\cos(180^\circ-\theta)=-\cos\theta}tan⁡(180∘−θ)=−tan⁡θ\boxed{\tan(180^\circ-\theta)=-\tan\theta}

180∘+θ180^\circ+\thetasin⁡(180∘+θ)=−sin⁡θ\boxed{\sin(180^\circ+\theta)=-\sin\theta}cos⁡(180∘+θ)=−cos⁡θ\boxed{\cos(180^\circ+\theta)=-\cos\theta}tan⁡(180∘+θ)=tan⁡θ\boxed{\tan(180^\circ+\theta)=\tan\theta}

360∘−θ360^\circ-\thetasin⁡(360∘−θ)=−sin⁡θ\boxed{\sin(360^\circ-\theta)=-\sin\theta}cos⁡(360∘−θ)=cos⁡θ\boxed{\cos(360^\circ-\theta)=\cos\theta}tan⁡(360∘−θ)=−tan⁡θ\boxed{\tan(360^\circ-\theta)=-\tan\theta}⭐ Exam trick

You don't need to memorise every formula separately if you understand:

1. Find the reference angle.

2. Determine the sign from the quadrant.

12. Sum and Difference of Angles ⭐⭐⭐

These are major Class 11 formulas.

Sinesin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\boxed{\sin(A+B)=\sin A\cos B+\cos A\sin B}sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\boxed{\sin(A-B)=\sin A\cos B-\cos A\sin B}

Cosinecos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\boxed{\cos(A+B)=\cos A\cos B-\sin A\sin B}cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\boxed{\cos(A-B)=\cos A\cos B+\sin A\sin B}

Notice the sign switches compared with sine.

Tangenttan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\boxed{\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}}tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\boxed{\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}}⭐⭐⭐ VERY IMPORTANT

These formulas are heavily used to derive standard-angle values and solve identities.

13. Multiple Angles ⭐⭐⭐Double AngleSinesin⁡2A=2sin⁡Acos⁡A\boxed{\sin2A=2\sin A\cos A}Cosinecos⁡2A=cos⁡2A−sin⁡2A\boxed{\cos2A=\cos^2A-\sin^2A}

Using sin⁡2A+cos⁡2A=1\sin^2A+\cos^2A=1:

cos⁡2A=2cos⁡2A−1\boxed{\cos2A=2\cos^2A-1}

or

cos⁡2A=1−2sin⁡2A\boxed{\cos2A=1-2\sin^2A}Tangenttan⁡2A=2tan⁡A1−tan⁡2A\boxed{\tan2A=\frac{2\tan A}{1-\tan^2A}}

14. Triple Angle ⭐⭐

Useful formulas:

sin⁡3A=3sin⁡A−4sin⁡3A\boxed{\sin3A=3\sin A-4\sin^3A}cos⁡3A=4cos⁡3A−3cos⁡A\boxed{\cos3A=4\cos^3A-3\cos A}tan⁡3A=3tan⁡A−tan⁡3A1−3tan⁡2A\boxed{\tan3A= \frac{3\tan A-\tan^3A} {1-3\tan^2A}}

These are more formula-heavy, so understand the double-angle formulas first.

15. General Solutions ⭐⭐⭐

This becomes important when solving trigonometric equations.

For example:

sin⁡x=0\sin x=0

The sine function is zero at:

0,π,2π,3π,…0,\pi,2\pi,3\pi,\ldots

So:

x=nπ,n∈Z\boxed{x=n\pi,\quad n\in\mathbb Z}

Ifcos⁡x=1\cos x=1

then:

x=2nπ\boxed{x=2n\pi}Ifcos⁡x=−1\cos x=-1

then:

x=(2n+1)π\boxed{x=(2n+1)\pi}Iftan⁡x=0\tan x=0

then:

x=nπ\boxed{x=n\pi}

16. Domains and Ranges ⭐⭐⭐

This is an important concept for understanding functions.

Sinesin⁡x\boxed{\sin x}

Domain:

R\mathbb R

Range:

[−1,1]\boxed{[-1,1]}

Cosinecos⁡x\boxed{\cos x}

Domain:

R\mathbb R

Range:

[−1,1]\boxed{[-1,1]}

Tangenttan⁡x\boxed{\tan x}

Domain:

x≠(2n+1)π2\boxed{x\neq\frac{(2n+1)\pi}{2}}

Range:

R\boxed{\mathbb R}

Why is tangent undefined there?

Because:

tan⁡x=sin⁡xcos⁡x\tan x=\frac{\sin x}{\cos x}

and cosine becomes zero.

Secantsec⁡x=1cos⁡x\sec x=\frac1{\cos x}

So it is undefined whenever:

cos⁡x=0\cos x=0

Range:

(−∞,−1]∪[1,∞)\boxed{(-\infty,-1]\cup[1,\infty)}

Cosecantcosec⁡x=1sin⁡x\cosec x=\frac1{\sin x}

Range:

(−∞,−1]∪[1,∞)\boxed{(-\infty,-1]\cup[1,\infty)}

Cotangentcot⁡x=cos⁡xsin⁡x\cot x=\frac{\cos x}{\sin x}

Range:

R\boxed{\mathbb R}

but it is undefined when:

sin⁡x=0\sin x=0

17. Trigonometric Graphs ⭐⭐

You should understand the basic shapes.

y=sin⁡xy=\sin x

• Maximum = 11

• Minimum = −1-1

• Period = 2π2\pi

y=cos⁡xy=\cos x

• Maximum = 11

• Minimum = −1-1

• Period = 2π2\pi

y=tan⁡xy=\tan x

• Period = π\pi

• No maximum/minimum

• Undefined at:

x=π2+nπx=\frac\pi2+n\pi⭐ Future importance

Graphs become extremely useful in:

• inverse trigonometry

• calculus

• differentiation

• integration

• oscillations/waves in physics

18. The Unit Circle — The Big Picture ⭐⭐⭐

If you understand this, the entire chapter becomes much easier.

Imagine a circle of radius:

1\boxed{1}

centred at the origin.

For an angle θ\theta, the point on the circle has coordinates:

(cos⁡θ,sin⁡θ)\boxed{(\cos\theta,\sin\theta)}

Therefore:

x=cos⁡θx=\cos\thetay=sin⁡θy=\sin\theta

And since:

x2+y2=1x^2+y^2=1

we get:

cos⁡2θ+sin⁡2θ=1\boxed{\cos^2\theta+\sin^2\theta=1}

That's where the fundamental identity comes from.

Then:

tan⁡θ=sin⁡θcos⁡θ\tan\theta=\frac{\sin\theta}{\cos\theta}

So:

tan⁡θ=yx\boxed{\tan\theta=\frac{y}{x}}

This is the conceptual foundation behind the signs, quadrants and standard values.

🧠 The Formula Sheet You Should Memorise

If you're doing last-minute revision, prioritise these.

Basictan⁡θ=sin⁡θcos⁡θ\boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}}cot⁡θ=cos⁡θsin⁡θ\boxed{\cot\theta=\frac{\cos\theta}{\sin\theta}}sec⁡θ=1cos⁡θ\boxed{\sec\theta=\frac1{\cos\theta}}cosec⁡θ=1sin⁡θ\boxed{\cosec\theta=\frac1{\sin\theta}}

Fundamental identities ⭐⭐⭐sin⁡2θ+cos⁡2θ=1\boxed{\sin^2\theta+\cos^2\theta=1}1+tan⁡2θ=sec⁡2θ\boxed{1+\tan^2\theta=\sec^2\theta}1+cot⁡2θ=cosec⁡2θ\boxed{1+\cot^2\theta=\cosec^2\theta}

Addition/Subtraction ⭐⭐⭐sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\boxed{\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B}cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\boxed{\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B}tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\boxed{\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}}tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\boxed{\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}}

Double angle ⭐⭐⭐sin⁡2A=2sin⁡Acos⁡A\boxed{\sin2A=2\sin A\cos A}cos⁡2A=2cos⁡2A−1\boxed{\cos2A=2\cos^2A-1}cos⁡2A=1−2sin⁡2A\boxed{\cos2A=1-2\sin^2A}cos⁡2A=cos⁡2A−sin⁡2A\boxed{\cos2A=\cos^2A-\sin^2A}tan⁡2A=2tan⁡A1−tan⁡2A\boxed{\tan2A=\frac{2\tan A}{1-\tan^2A}}

Periodicity ⭐⭐⭐sin⁡(x+2nπ)=sin⁡x\boxed{\sin(x+2n\pi)=\sin x}cos⁡(x+2nπ)=cos⁡x\boxed{\cos(x+2n\pi)=\cos x}tan⁡(x+nπ)=tan⁡x\boxed{\tan(x+n\pi)=\tan x}

🔥 What I Would Prioritise for Exams

For future Class 12 + Physics

The most valuable concepts to genuinely understand, rather than just memorise, are:

1. Unit circle

2. Radians

3. Quadrants and signs

4. Identities

5. Graphs/periodicity

6. Addition & subtraction formulas

Those concepts make later trigonometry feel much less like a giant formula sheet.