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Mathematics

Right Triangle Trigonometry

A High School & Early College Primer on SOH-CAH-TOA, Special Triangles, and Solving Triangles

Trigonometry stops a lot of students cold — not because it's hard, but because no one ever explained the ratios clearly before piling on formulas. If you have a test coming up, a geometry or pre-calc class that just hit SOH-CAH-TOA, or a kid who keeps asking "which side is opposite again?", this book gets you up to speed fast.

**TLDR: Right Triangle Trigonometry** covers exactly what the title says — right triangles, nothing more. You'll learn why sine, cosine, and tangent are just side ratios, how to use them to find missing sides and angles, the exact values you need for 30-60-90 and 45-45-90 special triangles, and how to fully solve a triangle from whatever information you're given. The final section works through real applications — angles of elevation and depression, building heights, river widths — the kind of word problems that show up on every geometry and pre-calculus exam.

This is a right triangle trig study guide for high school students in geometry, Algebra 2, or pre-calculus, and for early college students who need a clean, fast review. It is deliberately short: no unit circle, no trig identities, no Law of Sines. Those belong in the next book. This one builds a solid foundation first.

Each section leads with the one thing you must understand, then works through numbered examples with full solutions. Common mistakes are named and corrected inline.

Pick it up, read it once, do the examples — you'll be ready.

What you'll learn
  • Identify the opposite, adjacent, and hypotenuse sides relative to a chosen acute angle in a right triangle.
  • Define sine, cosine, and tangent as ratios of sides and remember them using SOH-CAH-TOA.
  • Compute exact trig values for the 30-60-90 and 45-45-90 special right triangles without a calculator.
  • Use inverse trig functions to find an unknown angle from two known sides.
  • Solve a right triangle (find all missing sides and angles) given enough information.
  • Set up and solve real-world problems involving angles of elevation, depression, and indirect measurement.
What's inside
  1. 1. Setting the Stage: Right Triangles, Sides, and Similarity
    Reviews the parts of a right triangle, fixes vocabulary for opposite/adjacent/hypotenuse, and explains why ratios of sides depend only on the angle.
  2. 2. The Three Ratios: Sine, Cosine, and Tangent
    Introduces sin, cos, and tan as side ratios, teaches SOH-CAH-TOA, and walks through finding a missing side using a known angle.
  3. 3. Special Right Triangles: 30-60-90 and 45-45-90
    Derives the exact side ratios of the two special triangles and uses them to find exact (non-decimal) trig values.
  4. 4. Going Backwards: Inverse Trig and Finding Angles
    Shows how to use arcsin, arccos, and arctan to recover an angle when you know two sides, including common calculator pitfalls.
  5. 5. Solving Right Triangles End-to-End
    Combines all the tools to fully solve right triangles given various starting information, with a checklist for which ratio to choose.
  6. 6. Applications: Elevation, Depression, and Indirect Measurement
    Applies right triangle trig to real problems — heights of buildings, distances across rivers, ramps, and navigation — using angles of elevation and depression.
Published by Solid State Press
Right Triangle Trigonometry cover
TLDR STUDY GUIDES

Right Triangle Trigonometry

A High School & Early College Primer on SOH-CAH-TOA, Special Triangles, and Solving Triangles
Solid State Press

Right Triangle Trigonometry

A High School & Early College Primer on SOH-CAH-TOA, Special Triangles, and Solving Triangles

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.

Who This Book Is For

If you are looking for a right triangle trig study guide for high school, you have found it. This book is for students in geometry or pre-calculus, anyone who bombed a quiz on trig ratios and needs a reset, and parents or tutors prepping a quick review session the night before a test.

The book covers everything you need to solve right triangles step by step: the sine, cosine, and tangent ratios built from first principles, the special right triangles — 30-60-90 and 45-45-90 — and inverse trig functions explained simply so you can recover an unknown angle from a ratio. Think of it as a focused SOH-CAH-TOA practice problems workbook that also explains the why behind every procedure. About 15 pages, no filler.

Read it straight through once — the sections build on each other. Work every example yourself before reading the solution. Then hit the problem set at the end, where you will find the trigonometry help for geometry class that ties it all together.

Contents

  1. 1 Setting the Stage: Right Triangles, Sides, and Similarity
  2. 2 The Three Ratios: Sine, Cosine, and Tangent
  3. 3 Special Right Triangles: 30-60-90 and 45-45-90
  4. 4 Going Backwards: Inverse Trig and Finding Angles
  5. 5 Solving Right Triangles End-to-End
  6. 6 Applications: Elevation, Depression, and Indirect Measurement
Chapter 1

Setting the Stage: Right Triangles, Sides, and Similarity

Every right triangle has one angle that measures exactly 90°, marked with a small square in diagrams. The side directly across from that right angle — always the longest side — is the hypotenuse. The other two sides are called legs. Everything in this book depends on keeping those three sides straight, so it is worth being precise from the start.

The trickier vocabulary involves the two legs, because their names change depending on which acute angle you are focused on. Pick either of the two non-right angles — call it angle $\theta$ (the Greek letter theta, used as a stand-in for any angle). The leg directly across from $\theta$ is the opposite side. The leg that forms one side of $\theta$ and is not the hypotenuse is the adjacent side. The hypotenuse is always the hypotenuse, but opposite and adjacent swap when you switch which angle you are calling $\theta$.

A common mistake is to label opposite and adjacent once and treat them as fixed properties of the triangle. They are not — they are properties of a side relative to a chosen angle. If you shift your attention from one acute angle to the other, the two legs trade roles.

Example. Triangle $ABC$ has a right angle at $C$. The sides are $BC = 3$, $AC = 4$, and $AB = 5$. Identify the hypotenuse, opposite, and adjacent sides relative to angle $A$.

Solution. The right angle is at $C$, so the hypotenuse is the side opposite that right angle: $AB = 5$. Now focus on angle $A$. The side directly across the triangle from $A$ is $BC = 3$ — that is the opposite side. The remaining leg, $AC = 4$, runs from $A$ to the right angle and is the adjacent side.

If you instead focus on angle $B$: the opposite side becomes $AC = 4$ and the adjacent side becomes $BC = 3$. The hypotenuse stays $AB = 5$.

The Pythagorean Theorem

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