Trigonometric Identities

Trigonometric identities are one of those topics that either clicks or doesn’t. Most resources make it worse. They dump formulas without context, skip the proofs that actually help you understand, or bury useful content under unnecessary complexity.

This guide takes a different approach.

Every identity includes its derivation. Every concept builds on the previous one. And the quick-reference tables at the end mean you won’t need to hunt through 70 pages when you just need to recall a formula during an exam.

Who This Is For

This ebook works best for:

• High school students preparing for board exams, JEE, or competitive tests
• Undergraduate students in engineering, physics, or mathematics courses
• Self-learners who want a structured reference that doesn’t assume prior knowledge
• Teachers and tutors who need a well-organized resource for lesson planning

If you can solve basic algebra problems, you have enough background to start. The guide builds from the unit circle and right triangle definitions up through Euler’s formula and hyperbolic functions.

What’s Inside

Part I: Foundations (Chapters 1-2)

The unit circle, degree-radian conversion, and all six trigonometric functions defined clearly. Includes the special angle value table you’ll reference constantly.

Part II: Fundamental Identities (Chapters 3-5)

Reciprocal, ratio, Pythagorean, and even-odd identities. Each Pythagorean identity includes a full proof so you understand where it comes from, not just what it is.

Part III: Sum, Difference, and Multiple Angle Identities (Chapters 6-9)

This is where most students struggle. The guide covers:

• Sum and difference formulas for sine, cosine, and tangent
• Double angle identities (all three forms of cos 2θ)
• Half angle formulas with sign determination
• Triple angle identities

Every formula includes worked examples showing exactly how to apply it.

Part IV: Product and Sum Identities (Chapters 10-11)

Product-to-sum and sum-to-product conversions. These show up in calculus integration problems constantly. The derivations here will save you from memorizing formulas blindly.

Part V: Cofunction and Periodicity (Chapters 12-13)

Complementary angles, supplementary angles, and the reference angle method for evaluating functions in any quadrant.

Part VI: Triangle Laws (Chapters 14-17)

Law of Sines, Law of Cosines, Law of Tangents, and area formulas including Heron’s formula. These chapters move beyond right triangles to solve any triangle problem.

Part VII: Advanced Topics (Chapters 18-20)

Inverse trigonometric functions, hyperbolic functions, and Euler’s formula. If you’re heading into calculus or complex analysis, this section bridges the gap.

Part VIII: Applications and Practice (Chapters 21-23)

Strategies for proving identities, derivatives and integrals of trig functions, and real-world applications in wave motion, harmonic oscillation, and AC circuits.

Quick Reference Tables

The final pages consolidate every major identity into scannable tables. Print these out. Keep them next to your desk during problem sets.

Specifications

• Pages: 73
• Format: PDF (works on any device)
• Includes: Worked examples, proofs, practice problems with answers, quick-reference tables
• Mathematical typesetting: Clean LaTeX formatting with clear diagrams