Trigonometry is one of those chapters that decides how comfortable you feel with the rest of your math syllabus — and later, with physics, engineering, and competitive exams. If you’ve ever searched for a trigonometry formulas PDF just before an exam, you already know the panic of scrolling through ten tabs trying to find one clean, correct list. This guide fixes that problem permanently.

Below, you’ll find every important trigonometric ratio, identity, and formula — organized by class level and exam relevance — along with a downloadable formula sheet you can save, print, and revise from anytime, even without internet access.

What Is Trigonometry? A Quick Refresher

Trigonometry comes from two Greek words — trigonon (triangle) and metron (measure). At its core, it studies the relationship between the angles and sides of a triangle, especially a right-angled triangle. These relationships are expressed through six functions: sine, cosine, tangent, cosecant, secant, and cotangent.

Once you understand these six ratios, almost every other formula in this guide is simply a variation or extension built on top of them. That’s why students preparing for CBSE boards, JEE Main, JEE Advanced, NEET, or SSC exams all rely on the same foundational set of identities — just applied differently depending on the difficulty level.

Why Keep a Trigonometry Formulas PDF Handy?

A well-organized, downloadable formula sheet saves you from three common problems:

• Formula confusion — mixing up sum and difference identities under exam pressure
• Time loss — flipping through an entire textbook to find one identity
• Inconsistent notes — formulas scattered across class notes, guidebooks, and online sources

Having a single, verified reference sheet means faster revision and fewer silly mistakes — especially in objective-based exams like JEE and SSC CGL, where speed matters as much as accuracy.

Basic Trigonometric Ratios (Foundation of Every Formula)

Every formula you’ll ever use in trigonometry traces back to these six basic ratios, defined using the sides of a right-angled triangle: Opposite (perpendicular), Adjacent (base), and Hypotenuse.

Ratio	Formula	Reciprocal Of
sin θ	Opposite / Hypotenuse	cosec θ
cos θ	Adjacent / Hypotenuse	sec θ
tan θ	Opposite / Adjacent	cot θ
cot θ	Adjacent / Opposite	tan θ
sec θ	Hypotenuse / Adjacent	cos θ
cosec θ	Hypotenuse / Opposite	sin θ

Source: National Council of Educational Research and Training (NCERT) — ncert.nic.in

This table alone answers most Class 10 board exam questions. Students moving into Class 11 and 12 build on these same six ratios to derive identities, compound angle formulas, and inverse functions.

Trigonometry Table: Standard Angle Values (0°, 30°, 45°, 60°, 90°)

This is one of the most searched sections in any trigonometric reference sheet, since these exact values appear repeatedly in board exams and competitive tests.

Angle →	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	Not Defined
cot θ	Not Defined	√3	1	1/√3	0
sec θ	1	2/√3	√2	2	Not Defined
cosec θ	Not Defined	2	√2	2/√3	1

Source: CBSE Academic Curriculum, NCERT Mathematics Textbook — cbseacademic.nic.in

Memory tip: Notice that sin values increase from 0 to 1 as the angle rises from 0° to 90°, while cos values do the exact opposite. Once this pattern clicks, you rarely need to memorize the table by rote.

Extended Trigonometry Table: Values for 15°, 18°, 36°, and 75°

Beyond the five standard angles, JEE and Olympiad-level questions often use a few additional “special” angles derived using half-angle and sum-difference formulas. Keeping these handy saves derivation time during exams.

Angle	sin θ	cos θ	tan θ
15°	(√6 − √2)/4	(√6 + √2)/4	2 − √3
18°	(√5 − 1)/4	√(10 + 2√5)/4	—
36°	√(10 − 2√5)/4	(√5 + 1)/4	—
75°	(√6 + √2)/4	(√6 − √2)/4	2 + √3

Source: NCERT Exemplar Problems, Class 11 Mathematics — ncert.nic.in

Sign of Trigonometric Functions in Each Quadrant (ASTC Rule)

Knowing which functions are positive or negative in a given quadrant is essential for solving equations and simplifying expressions correctly. This is remembered using the “ASTC” rule — All, Sine, Tangent, Cosine.

Quadrant	Angle Range	Positive Functions	Negative Functions
I	0° – 90°	All (sin, cos, tan, and reciprocals)	None
II	90° – 180°	sin, cosec	cos, tan, sec, cot
III	180° – 270°	tan, cot	sin, cos, sec, cosec
IV	270° – 360°	cos, sec	sin, tan, cot, cosec

Source: NCERT Class 11 Mathematics, Chapter — Trigonometric Functions

A quick way to remember ASTC: “Add Sugar To Coffee” — each letter marks which functions stay positive in that quadrant, moving anticlockwise from quadrant I.

Allied Angle (Co-Function) Formulas

Allied angles are angles that are integer multiples of 90° added to or subtracted from a given angle θ. These formulas let you convert an unfamiliar angle into a known reference angle, and they appear constantly in simplification and proof-based questions.

Angle Form	sin	cos	tan
(90° − θ)	cos θ	sin θ	cot θ
(90° + θ)	cos θ	−sin θ	−cot θ
(180° − θ)	sin θ	−cos θ	−tan θ
(180° + θ)	−sin θ	−cos θ	tan θ
(270° − θ)	−cos θ	−sin θ	cot θ
(270° + θ)	−cos θ	sin θ	−cot θ
(360° − θ)	−sin θ	cos θ	−tan θ
(360° + θ)	sin θ	cos θ	tan θ

Source: NCERT Class 11 Mathematics Textbook, Chapter 3

Quick rule: For 90° and 270°, the function name changes (sin ↔ cos, tan ↔ cot). For 180° and 360°, the function name stays the same — only the sign changes based on the quadrant (use the ASTC table above to check the sign).

Negative Angle (Even-Odd) Formulas

These identities tell you what happens when the angle itself is negative — a small detail that trips up a lot of students in simplification problems.

Function	Identity	Type
sin(−θ)	−sin θ	Odd
cos(−θ)	cos θ	Even
tan(−θ)	−tan θ	Odd
cot(−θ)	−cot θ	Odd
sec(−θ)	sec θ	Even
cosec(−θ)	−cosec θ	Odd

Source: NCERT Class 11 Mathematics, Chapter — Trigonometric Functions

Periodicity of Trigonometric Functions

Every trigonometric function repeats its values after a fixed interval, called its period. This concept is what makes trigonometric equations have infinite solutions and is frequently tested in Class 11 and JEE.

Function	Period
sin θ, cos θ, sec θ, cosec θ	2π (360°)
tan θ, cot θ	π (180°)

Source: NCERT Class 11 Mathematics Textbook

Complete Trigonometry Formulas PDF: Quick-Reference Chart

At this point, you’ve covered ratios, standard angles, extended angles, quadrant signs, allied angles, negative-angle rules, and periodicity — which together make up the first half of any complete trigonometry formulas PDF. Before moving into identities and compound-angle formulas, it helps to pause and see how these categories fit together, so you’re not just memorizing isolated tables.

Formula Category	Number of Key Formulas	Typically Tested In
Basic Ratios & Reciprocals	6	Class 10, 11
Standard & Extended Angle Values	9 angles	Class 10, 11, JEE
Quadrant Signs (ASTC)	4 quadrants	Class 11
Allied Angle Formulas	8	Class 11, JEE
Negative Angle Formulas	6	Class 11
Periodicity Rules	2	Class 11, JEE

Source: Compiled from NCERT Class 10 & 11 Mathematics Textbooks — ncert.nic.in

Why a Trigonometry Formulas PDF Chart Works Better Than Scattered Notes

Seeing every category in one chart, rather than as separate scattered notes, makes it far easier to spot which formulas you’ve actually memorized and which ones still need practice. Most students who revise from a single, well-organized sheet report fewer mix-ups between similar-looking rules — such as confusing allied angle sign changes with quadrant sign changes — simply because both appear side by side instead of in different notebooks or textbook chapters.

The remaining categories — identities, compound-angle formulas, product-to-sum conversions, and inverse functions — build directly on the six categories above, which is why getting this first half solid before moving forward makes the rest of the chapter noticeably easier.

Trigonometric Identities: Pythagorean & Reciprocal Relations

These identities form the backbone of Class 11 trigonometry and are used constantly to simplify expressions and prove equations.

Identity Type	Formula
Pythagorean Identity 1	sin²θ + cos²θ = 1
Pythagorean Identity 2	1 + tan²θ = sec²θ
Pythagorean Identity 3	1 + cot²θ = cosec²θ
Reciprocal Identity	sin θ = 1 / cosec θ
Reciprocal Identity	cos θ = 1 / sec θ
Quotient Identity	tan θ = sin θ / cos θ
Quotient Identity	cot θ = cos θ / sin θ

Source: NCERT Class 11 Mathematics, Chapter — Trigonometric Functions

These identities are derived directly from the Pythagoras theorem applied to a right-angled triangle, which is why they hold true for every value of θ within the function’s domain.

Sum, Difference, Double & Half-Angle Formulas

This section is where most students lean heavily on a saved reference sheet, since these formulas are lengthy and easy to misremember under time pressure.

Sum and Difference Formulas

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

Double Angle Formulas

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
• tan 2θ = 2 tan θ / (1 − tan²θ)

Triple Angle Formulas

• sin 3θ = 3 sin θ − 4 sin³θ
• cos 3θ = 4 cos³θ − 3 cos θ
• tan 3θ = (3 tan θ − tan³θ) / (1 − 3 tan²θ)

Half-Angle Formulas

• sin(θ/2) = ±√[(1 − cos θ)/2]
• cos(θ/2) = ±√[(1 + cos θ)/2]
• tan(θ/2) = ±√[(1 − cos θ)/(1 + cos θ)]

These compound-angle formulas are heavily tested in JEE Main and JEE Advanced, where questions often require combining two or three identities in a single problem — another reason a consolidated formula sheet is worth keeping on your device.

Product-to-Sum and Sum-to-Product Formulas

Once you’re comfortable with the sum, difference, and double-angle formulas, the next logical step is product-to-sum and sum-to-product identities. These appear less frequently in Class 10 and 11 syllabi but show up often in JEE Advanced and higher-level engineering mathematics, where converting a product of trigonometric terms into a sum (or vice versa) simplifies integration and equation-solving.

Conversion Type	Formula
Product to Sum	2 sin A cos B = sin(A+B) + sin(A−B)
Product to Sum	2 cos A cos B = cos(A−B) + cos(A+B)
Product to Sum	2 sin A sin B = cos(A−B) − cos(A+B)
Sum to Product	sin A + sin B = 2 sin[(A+B)/2] cos[(A−B)/2]
Sum to Product	cos A + cos B = 2 cos[(A+B)/2] cos[(A−B)/2]

Source: NCERT Class 11 Mathematics, Chapter — Trigonometric Functions

These conversions are particularly useful in calculus-based problems, where integrating a product of sine and cosine terms directly is difficult, but integrating their sum form is straightforward. Engineering aspirants preparing for JEE Advanced should treat this table as a priority section, since it frequently appears in integration and equation-solving questions.

Trigonometric Equations and General Solutions

Beyond identities, trigonometry also involves solving equations where an unknown angle satisfies a given trigonometric condition. These are called trigonometric equations, and unlike algebraic equations, they usually have infinite solutions due to the periodic nature of trigonometric functions.

Equation Type	General Solution
sin θ = sin α	θ = nπ + (−1)ⁿα, n ∈ Z
cos θ = cos α	θ = 2nπ ± α, n ∈ Z
tan θ = tan α	θ = nπ + α, n ∈ Z

Here, “n” represents any integer, which accounts for the fact that sine, cosine, and tangent repeat their values at regular intervals (periodicity). This concept is introduced in Class 11 and becomes especially important for JEE-level problem-solving, where a single trigonometric equation can have multiple valid answers within a given range.

Inverse Trigonometric Formulas

Inverse trigonometric functions reverse the usual ratio-to-angle relationship — instead of finding a ratio from an angle, you find the angle from a given ratio. These are introduced in Class 12 and are a scoring topic if practiced well.

Function	Domain	Range
sin⁻¹x	[−1, 1]	[−π/2, π/2]
cos⁻¹x	[−1, 1]	[0, π]
tan⁻¹x	(−∞, ∞)	(−π/2, π/2)

Source: NCERT Class 12 Mathematics, Chapter — Inverse Trigonometric Functions

Common inverse identities include:

• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• sin⁻¹(−x) = −sin⁻¹x

How to Use This Formula Sheet for Revision

Getting the formulas is only half the job — using them efficiently matters more. Here’s a simple approach that toppers use:

1. Print the trigonometry formulas PDF and stick it somewhere visible — your study desk or the inside cover of your notebook.
2. Group formulas by type (ratios, identities, compound angles) instead of memorizing them as one giant list.
3. Solve 5–10 problems daily applying at least one formula from each category.
4. Revisit the standard angle table weekly until it becomes automatic recall.
5. Use the PDF the night before exams as a rapid-fire revision tool rather than reading full chapters again.

Real-World Applications of Trigonometry Formulas

Trigonometry isn’t just an exam topic — it has practical relevance that makes the formulas easier to remember once you see them in context:

• Architecture and construction — calculating heights, roof angles, and structural load distribution
• Navigation and astronomy — determining distances between celestial bodies and ship positioning at sea
• Engineering — signal processing, wave mechanics, and electrical circuit analysis rely on sine and cosine functions
• Geography — triangulation methods used in mapping distances between landmarks
• Computer graphics — rotation, animation, and 3D modeling all use trigonometric transformations

Understanding these applications makes the formula list feel less like abstract memorization and more like a practical toolkit.

Class-Wise Relevance: What to Focus On

Class Level	Key Focus Areas
Class 10	Basic ratios, Pythagorean identity, heights & distances
Class 11	Trigonometric functions, identities, compound angle formulas, graphs
Class 12	Inverse trigonometric functions, equations, application in calculus
JEE/NEET/SSC	Multiple angle formulas, transformation formulas, equations with general solutions

Source: CBSE Curriculum Framework — cbse.gov.in

This breakdown helps you know exactly which section of the formula guide to prioritize based on your current class or target exam, rather than trying to memorize everything at once.

Formula Sheet vs Full Textbook: Which Should You Revise From?

Many students wonder whether it’s better to revise from a compact formula sheet or go back to the full textbook chapter every time. Both have a place in your study routine, but they serve different purposes.

Factor	Formula Sheet (PDF)	Full Textbook Chapter
Best for	Quick revision, last-minute recall	First-time learning, understanding derivations
Time required	5–10 minutes	45–60 minutes
Ideal usage stage	Week before exams, daily practice warm-up	Initial concept building
Portability	High — works offline, fits on one page	Low — requires carrying the full book
Depth of explanation	Minimal, formula-focused	Detailed, includes proofs and solved examples

The most effective students don’t choose one over the other — they learn concepts from the textbook first, then switch to a condensed reference sheet for ongoing revision. Relying only on a formula list without ever understanding the derivation tends to create shaky recall under exam pressure, while relying only on the textbook wastes valuable revision time closer to exams.

Common Mistakes Students Make With Trigonometry Formulas

• Confusing sin(A+B) with sin A + sin B — these are not the same
• Forgetting the ± sign in half-angle formulas depending on the quadrant
• Mixing up domain and range for inverse trigonometric functions
• Applying degree-based values in radian-based questions without converting
• Skipping the derivation logic and memorizing formulas as isolated symbols, which makes recall harder under pressure

Avoiding these errors comes down to practicing derivations occasionally, not just formulas — even five minutes of writing out how sin(A+B) is derived cements the formula far better than repetition alone.

Summary: Key Takeaways From This Trigonometry Formulas PDF

• The six basic trigonometric ratios (sin, cos, tan, cot, sec, cosec) are the foundation for every other formula.
• The standard angle table (0°, 30°, 45°, 60°, 90°) is essential and frequently tested across all classes.
• Pythagorean and reciprocal identities simplify complex trigonometric expressions.
• Sum, difference, double, and half-angle formulas are critical for JEE and advanced problem-solving.
• Inverse trigonometric functions matter most for Class 12 and calculus-based applications.
• Keeping a single, verified formula sheet for revision saves significant exam preparation time.

Frequently Asked Questions (FAQs)

Conclusion

Trigonometry rewards consistency more than raw memorization. Once the six basic ratios and the standard angle table are second nature, every other formula in this trigonometry formulas PDF becomes an extension of concepts you already understand rather than a separate list to cram. Keep this guide as your primary reference sheet, revisit it weekly, and pair it with regular problem-solving — that combination is what actually turns formula knowledge into exam-ready recall, whether you’re preparing for board exams, JEE, NEET, or SSC-level competitive tests.

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