SOLID STATE PRESS
← Back to catalog
Graphing Sine and Cosine cover
Coming soon
Coming soon to Amazon
This title is in our publishing queue.
Browse available titles
Mathematics

Graphing Sine and Cosine

A High School and Early College Primer on Amplitude, Period, and Phase Shift

Sine and cosine graphs show up on every precalculus exam, every AP Calculus free-response section, and every trigonometry quiz — and they trip students up for one reason: too many moving parts at once. Amplitude, period, phase shift, vertical shift, and the sign traps inside the formula all stack on top of each other, and most textbooks explain each piece separately without ever showing how they work together.

**TLDR: Graphing Sine and Cosine** fixes that. In under 20 pages, you get a clean, step-by-step walkthrough of the full transformation formula $y = A\sin(B(x - C)) + D$, starting from where sine and cosine actually come from (the unit circle) and building to a five-step procedure you can apply to any sinusoidal function on sight. The guide covers amplitude and vertical shift, how to graph sine and cosine functions with changed periods, the phase shift sign-error trap that costs students points on nearly every exam, and how to reverse the process and write an equation from a graph.

This guide is written for high school students in precalculus or trigonometry, early college students in College Algebra or Calculus I, and parents or tutors who need a fast, honest refresher before a session. It skips the filler and gets to what you actually need: clear definitions, worked examples with real numbers, and a repeatable process.

If a trig graphing test is coming up, pick this up today and walk in prepared.

What you'll learn
  • Sketch the parent graphs of y = sin(x) and y = cos(x) from memory and identify their key features.
  • Interpret each constant in y = A sin(B(x - C)) + D and explain how it transforms the graph.
  • Find the amplitude, period, phase shift, and vertical shift of any sinusoidal function.
  • Graph a transformed sine or cosine curve by locating its midline and five key points.
  • Write an equation for a sinusoidal graph given its picture or verbal description.
  • Recognize and avoid common mistakes involving the sign of the phase shift and the role of B.
What's inside
  1. 1. The Unit Circle Meets the Graph: Where Sine and Cosine Come From
    Connects the unit circle definitions of sine and cosine to the shape of their graphs and establishes the parent curves.
  2. 2. Amplitude and Vertical Shift: Stretching and Sliding Up and Down
    Explains how the constants A and D in y = A sin(x) + D control the height and midline of the graph.
  3. 3. Period and Horizontal Stretch: What B Does
    Shows how the coefficient B inside the function changes the period and how to compute the new period correctly.
  4. 4. Phase Shift: Sliding the Graph Left and Right
    Unpacks the phase shift in y = A sin(B(x - C)) + D, including the common sign-error trap and how to factor B out correctly.
  5. 5. Putting It All Together: Graphing in Five Steps
    A repeatable procedure for graphing any y = A sin(B(x - C)) + D or cosine analog, with two full worked examples.
  6. 6. Reading the Graph Backward: Writing Equations and Real Uses
    Teaches how to recover an equation from a sinusoidal graph and shows where these curves appear in physics, biology, and signal processing.
Published by Solid State Press
Graphing Sine and Cosine cover
TLDR STUDY GUIDES

Graphing Sine and Cosine

A High School and Early College Primer on Amplitude, Period, and Phase Shift
Solid State Press

Graphing Sine and Cosine

A High School and Early College Primer on Amplitude, Period, and Phase Shift

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 sitting in Precalculus or Trigonometry and your teacher just introduced sinusoidal graphs, this book is for you. It's also for the student pulling up a graphing sinusoidal functions worksheet the night before a test and realizing they never quite understood where the numbers come from — and for the parent or tutor who needs a fast, honest refresher.

This precalculus sine cosine study guide walks through every core idea: how to graph sine and cosine functions from scratch, what amplitude, period, and phase shift mean and how each one changes the picture, and how vertical shifts fit in. Think of it as amplitude, period, and phase shift explained in the order that actually makes sense, in about 15 focused pages with no filler.

Start at page one and read straight through. Work every example as you go — don't skip them. The problem set at the end is your check: if you can do those, you're ready for precalc test prep, trigonometry graphs included, and the phase shift and period trig quick review is built right in.

Contents

  1. 1 The Unit Circle Meets the Graph: Where Sine and Cosine Come From
  2. 2 Amplitude and Vertical Shift: Stretching and Sliding Up and Down
  3. 3 Period and Horizontal Stretch: What B Does
  4. 4 Phase Shift: Sliding the Graph Left and Right
  5. 5 Putting It All Together: Graphing in Five Steps
  6. 6 Reading the Graph Backward: Writing Equations and Real Uses
Chapter 1

The Unit Circle Meets the Graph: Where Sine and Cosine Come From

Imagine a point crawling around a circle of radius 1, centered at the origin. That image is the entire source of sine and cosine.

The unit circle is that circle — radius exactly 1, center at $(0, 0)$. Any point on it can be described by an angle $\theta$ (the Greek letter theta) measured counterclockwise from the positive $x$-axis. The two coordinates of that point are defined to be the cosine and sine of $\theta$:

$(\cos\theta,\ \sin\theta)$

So cosine is the $x$-coordinate of the point, and sine is the $y$-coordinate. That's the whole definition. Everything else follows from it.

Radians: the natural unit for this

Angles here are measured in radians, not degrees. One radian is the angle swept out when the arc length along the circle equals the radius. Since the unit circle has radius 1, a full trip around the circle covers an arc length of $2\pi$, so a full revolution equals $2\pi$ radians. The conversion you need:

$360° = 2\pi \text{ rad}, \qquad 180° = \pi \text{ rad}$

The key landmarks: $0,\ \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ 2\pi$ correspond to the rightmost point, the top, the leftmost point, the bottom, and back to the start. Memorize those five positions — they become the five key points for every graph in this book.

Tracing the coordinates as the point moves

Watch what happens to the $y$-coordinate (sine) as the point travels counterclockwise:

  • At $\theta = 0$: the point is at $(1, 0)$, so $\sin 0 = 0$.
  • At $\theta = \frac{\pi}{2}$: the point is at $(0, 1)$, so $\sin\frac{\pi}{2} = 1$.
  • At $\theta = \pi$: the point is at $(-1, 0)$, so $\sin\pi = 0$.
  • At $\theta = \frac{3\pi}{2}$: the point is at $(0, -1)$, so $\sin\frac{3\pi}{2} = -1$.
  • At $\theta = 2\pi$: back to $(1, 0)$, so $\sin 2\pi = 0$.
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