SOLID STATE PRESS
← Back to catalog
Sum, Difference, and Double-Angle Identities cover
Coming soon
Coming soon to Amazon
This title is in our publishing queue.
Browse available titles
Mathematics

Sum, Difference, and Double-Angle Identities

Sin, Cos, and Tan of Sums and Double Angles — A TLDR Primer

Trig identities have a reputation for being the section where students fall behind and never quite catch up. The formulas look like alphabet soup, the derivations feel endless, and by the time an exam arrives, most students are guessing which formula goes where.

This TLDR guide cuts straight to what matters. Short by design, you'll understand the sum and difference identities for sine, cosine, and tangent — where they come from, how to read them, and how to apply them to find exact values for angles like 15° and 75° that the unit circle doesn't hand you. You'll see how the double-angle identities follow directly from the sum formulas (no memorization required once you see the pattern), work through the three forms of cos(2θ) and when each one is useful, and solve the kinds of problems that show up on precalculus and trigonometry exams.

This book is written for high school students in precalculus or trigonometry, early college students in a trig or STEM survey course, and parents or tutors who need to get up to speed fast. Every key term is defined the first time it appears. Every identity comes with a worked example. The final section shows where these formulas reappear — in calculus integrals, physics wave equations, and half-angle identities — so the time you invest here pays forward.

If you have a test this week or a concept that still isn't clicking, pick this up and read it once.

What you'll learn
  • State the sum and difference identities for sine, cosine, and tangent from memory
  • Derive the double-angle identities from the sum identities
  • Use the identities to find exact values of trig expressions like sin(75°) or cos(2θ)
  • Simplify and verify trig expressions involving sums, differences, and double angles
  • Solve basic trig equations that require a double-angle or sum identity
What's inside
  1. 1. Why We Need These Identities
    Motivates the identities by showing what they let you do that the unit circle alone can't.
  2. 2. The Sum and Difference Identities
    Presents the six core sum/difference formulas for sin, cos, and tan, with a geometric sketch of why cos(A−B) is true.
  3. 3. Using Sum and Difference Identities
    Worked examples: exact values of angles like 15° and 75°, simplifying expressions, and verifying identities.
  4. 4. The Double-Angle Identities
    Derives sin(2θ), cos(2θ), and tan(2θ) from the sum identities and explains the three forms of cos(2θ).
  5. 5. Working Problems with Double Angles
    Applies double-angle identities to find values given partial info, simplify expressions, and solve equations.
  6. 6. Where This Shows Up Next
    Brief tour of where these identities reappear: calculus integrals, physics waves, and half-angle/product-to-sum formulas.
Published by Solid State Press
Sum, Difference, and Double-Angle Identities cover
TLDR STUDY GUIDES

Sum, Difference, and Double-Angle Identities

Sin, Cos, and Tan of Sums and Double Angles — A TLDR Primer
Solid State Press

Sum, Difference, and Double-Angle Identities

Sin, Cos, and Tan of Sums and Double Angles — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 Why We Need These Identities
  2. 2 The Sum and Difference Identities
  3. 3 Using Sum and Difference Identities
  4. 4 The Double-Angle Identities
  5. 5 Working Problems with Double Angles
  6. 6 Where This Shows Up Next
Chapter 1

Why We Need These Identities

Your unit circle lets you find the sine and cosine of a tidy list of angles: 0°, 30°, 45°, 60°, 90°, and their cousins in other quadrants. That list is genuinely useful. But it is also genuinely short. The moment an exam or a physics problem hands you sin(75°) or cos(195°), the unit circle goes silent.

This is exactly the gap that the sum, difference, and double-angle identities fill.

An identity is an equation that is true for every value of the variable — not just one or two solutions, but always. Compare that to an ordinary equation like $2\sin\theta = 1$, which is only satisfied by specific values of $\theta$. The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ is a classic example of an identity: plug in any angle at all and both sides match. The identities you will learn in this book have that same flavor — they hold universally, which is what makes them powerful rewriting tools rather than problems to solve.

What the unit circle gives you

The unit circle is the circle of radius 1 centered at the origin. For any angle $\theta$ measured from the positive $x$-axis, the point on the circle is $(\cos\theta,\, \sin\theta)$. That definition is the foundation of all of trigonometry.

From the unit circle you can read off exact values at the standard angles. For instance:

$\cos 60° = \frac{1}{2}, \qquad \sin 45° = \frac{\sqrt{2}}{2}, \qquad \tan 30° = \frac{1}{\sqrt{3}}$

These are exact — no decimals, no rounding. Exact values matter in mathematics because they let you simplify, compare, and build on results without accumulating numerical error.

About This Book

If you are a high school student who needs a trig identities study guide before a precalculus test, a student in an early college trig course, or a parent helping a teenager prepare for an exam, this book was written for you. It also works as a fast refresher for anyone who covered these formulas once and needs them back in working order.

This guide covers the sum and difference formulas from precalculus — sin, cos, and tan of combined angles — along with the double-angle identities and their derivations. Think of it as a trigonometry identities quick reference guide that actually explains the reasoning, not just the formulas. The sin and cos identities are presented for beginners without assuming prior mastery, and every major formula comes with a worked example. A concise overview with no filler.

Read straight through once to build the framework, work each example as you reach it, then use the double-angle identities practice problems at the end to confirm your understanding. For precalc test prep, students find that order makes the material stick.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon