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Mathematics

Right Triangles and Trigonometry

A High School & Early College Primer

Trigonometry shows up on the SAT, in geometry class, and on the first exam of precalculus — and a lot of students hit it without a clear foundation. If the Pythagorean theorem feels fuzzy, sine and cosine are just letters, and 30-60-90 triangles seem like magic, this guide is the fix.

**TLDR: Right Triangles and Trigonometry** covers everything a high school or early college student needs to get comfortable with right-triangle math — fast. The book walks through the Pythagorean theorem with worked numbers, derives the side ratios for the two special right triangles that appear constantly on exams, and builds SOH-CAH-TOA from scratch so the ratios actually make sense instead of just being memorized. From there it covers solving complete triangles, setting up angles-of-elevation and angles-of-depression problems, and closes with a clear-eyed preview of where trig goes next: the unit circle, reciprocal ratios, and the Law of Sines.

This is a focused primer, not a textbook. It runs about 15 pages — long enough to be thorough, short enough to read in one sitting before a test. Every section leads with the one idea you need to take away, names the most common student mistakes, and backs every rule with a concrete worked example. Parents helping a kid through geometry night, tutors prepping a session, or students who need a right triangle trig study guide that gets to the point will all find exactly what they need here.

Pick it up, read it once, and walk into your next exam oriented.

What you'll learn
  • Apply the Pythagorean theorem and recognize common right-triangle patterns (3-4-5, 5-12-13, 45-45-90, 30-60-90).
  • Define sine, cosine, and tangent as ratios of sides in a right triangle and use SOH-CAH-TOA fluently.
  • Solve for unknown sides and angles using trig ratios and inverse trig functions.
  • Set up and solve real-world problems involving angles of elevation and depression.
  • Understand reciprocal ratios (cosecant, secant, cotangent) and the basic identities that connect them.
What's inside
  1. 1. Right Triangles and the Pythagorean Theorem
    Introduces right triangles, the names of their sides, and the Pythagorean theorem with worked examples.
  2. 2. Special Right Triangles: 45-45-90 and 30-60-90
    Derives and applies the side ratios for the two special right triangles that appear constantly in geometry and trig.
  3. 3. Sine, Cosine, and Tangent: SOH-CAH-TOA
    Defines the three primary trig ratios as side ratios in a right triangle and shows how to compute them.
  4. 4. Solving Right Triangles: Finding Sides and Angles
    Uses trig ratios and inverse trig functions to find unknown sides and angles, completing a full triangle.
  5. 5. Applications: Angles of Elevation and Depression
    Translates real-world height, distance, and ramp problems into right-triangle setups and solves them.
  6. 6. Reciprocal Ratios and Where Trig Goes Next
    Introduces cosecant, secant, cotangent, the Pythagorean identity, and previews the unit circle and Law of Sines/Cosines.
Published by Solid State Press
Right Triangles and Trigonometry cover
TLDR STUDY GUIDES

Right Triangles and Trigonometry

A High School & Early College Primer
Solid State Press

Right Triangles and Trigonometry

A High School & Early College 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.

Who This Book Is For

If you're a high school student working through a geometry or precalculus unit on triangles, this right triangle trig study guide for high school is written for you. It also fits anyone running short on time before the SAT — this doubles as an SAT math trigonometry prep book — or a parent helping a student review before a test.

The book covers the Pythagorean theorem (with practice problems built in), the special right triangles 45-90 and 30-60 guide, and SOH CAH TOA explained for beginners who have never seen a trig ratio before. From there it moves into the sine cosine tangent skills that function as a workbook for students solving real triangle problems, then closes with angles of elevation, depression, and a preview of where trig goes next in precalculus. These geometry trig ratios get a quick review treatment: about 15 pages, no filler.

Read straight through in order. Work every example yourself before reading the solution, then tackle the problem set at the end to confirm you've got it.

Contents

  1. 1 Right Triangles and the Pythagorean Theorem
  2. 2 Special Right Triangles: 45-45-90 and 30-60-90
  3. 3 Sine, Cosine, and Tangent: SOH-CAH-TOA
  4. 4 Solving Right Triangles: Finding Sides and Angles
  5. 5 Applications: Angles of Elevation and Depression
  6. 6 Reciprocal Ratios and Where Trig Goes Next
Chapter 1

Right Triangles and the Pythagorean Theorem

A right triangle is a triangle that contains one 90-degree angle, marked with a small square in diagrams. That single constraint — one right angle — locks in a precise relationship among the three sides, and exploiting that relationship is what this entire book is about.

Naming the Sides

The sides of a right triangle have specific names, and using them correctly matters because the formulas ahead depend on them.

The two sides that form the right angle are called legs. The side opposite the right angle — always the longest side — is the hypotenuse. Every right triangle has exactly two legs and one hypotenuse. A common slip is to call the hypotenuse whichever side looks longest in a sketch; the correct rule is purely positional: the hypotenuse is always across from the 90-degree angle, no exceptions.

When you label a right triangle, you'll often see the legs called $a$ and $b$ and the hypotenuse called $c$. That's the convention used here and in most textbooks.

The Pythagorean Theorem

If you know any two sides of a right triangle, you can always find the third. The rule that makes this possible is the Pythagorean theorem:

$a^2 + b^2 = c^2$

In words: the sum of the squares of the two legs equals the square of the hypotenuse. This is one of the most-used equations in mathematics — it shows up in geometry, physics, engineering, and later sections of this book. Knowing it cold is non-negotiable.

A word on what the theorem is not saying: it does not say the sides add up to the hypotenuse. $a + b \neq c$. The relationship is between the squares of the sides. That distinction trips up students on exams regularly.

Example. A right triangle has legs of length 6 and 8. Find the hypotenuse.

Solution. Plug into $a^2 + b^2 = c^2$: $6^2 + 8^2 = c^2$ $36 + 64 = c^2$ $100 = c^2$ $c = \sqrt{100} = 10$

The hypotenuse is 10.

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