Ratios and Unit Rates
Tables, diagrams, and fair comparisons
Math workbook
Right-Triangle Trigonometry: Sine, cosine, tangent and indirect measurement
Grades 9–12Level 4
About this book
Choose the right ratio, solve the triangle, and justify your setup.
This workbook teaches right-triangle trigonometry one clear step at a time. Learners see why sine, cosine and tangent belong to an angle, using similar triangles. They then find missing sides, including the tricky case where the unknown is in the denominator, and use inverse sine, cosine and tangent to find and check angles. The final module turns trees, towers and ramps into right triangles, with eye-height offsets and measurement error. Every module follows the same seven steps, with spaced review, an exit check and a full explanatory answer key for adults.
- Name opposite, adjacent and hypotenuse from any angle
- Solve for a side, even with the unknown on the bottom
- Find and verify angles with sin⁻¹, cos⁻¹ and tan⁻¹
- Model heights and distances with elevation and depression
- Answer key with reasons, common errors and reteach cues
For learners in grades 9–12 who can use the Pythagorean theorem, similar triangles and proportions.
Look inside
What this book covers
Covered
- In scope: acute angles in right triangles, measured in degrees.
- Pages 53–54 add exact 30°–60°–90° and 45°–45°–90° triangles.
Not covered
Not in scope: the law of sines and cosines, radians, the unit circle, trigonometric graphs and identities.
Before you start
You need first: the Pythagorean theorem, similar triangles (Geometry Proofs: Build a Logical Argument), proportions (Ratios and Unit Rates) and solving one-variable equations (Expressions and One-Variable Equations).
Things to have
Tools: a pencil, a ruler and a scientific calculator in degree mode.
Notice
Pictures in this book are simplified learning models. Check answers with the explanations at the back of the book.
The steps in this book
- Step 1: Readiness Check
Check the skills you need before you begin. If something feels tricky, an adult can help you start in the right place.
- Step 2: Concept
Read a short, clear explanation of the new idea and the words that go with it.
- Step 3: Worked Example
Watch the idea being used, one step at a time, in a worked example.
- Step 4: Guided Practice
Practice with prompts and help cues. The help gets smaller as you get stronger.
- Step 5: Independent Practice
Practice on your own with different kinds of questions, and explain your thinking.
- Step 6: Review
Look back at earlier learning so it stays strong, then check where to go next.
- Step 7: Answers and Explanations
Check your answers. Each answer explains why, names a common mistake, and shows how to fix it.
What is in the book
The modules
- Module 1: Ratios belong to an anglePages 5–14
- Module 2: Find a missing sidePages 15–24
- Module 3: Find and verify an anglePages 25–34
- Module 4: Model an indirect measurementPages 35–44
Contents
- Readiness check, pp. 3–4. Before Module 1. Its table tells you where to start.
- Modules, pp. 5–44
- Checkpoints, pp. 14, 24, 34, 44. The last page of each module.
- Mixed review, pp. 45–48. Questions from across the book, mixed together.
- Exit check, pp. 49–50. A last check across the book.
- Reflect and repair, p. 51
- Try again, p. 52. Fresh questions like those in the exit check.
- Further practice, pp. 53–54. More pages of step 5, Independent Practice.
- Extra practice, pp. 55–57
- Answers and explanations, pp. 58–66. Notes in this part are headed “Why”, “Common mistake” and “Help”.
Other books it mentions
Before this book
Expressions and One-Variable Equations
Variables, equivalence, and balance
Geometry Proofs: Build a Logical Argument
Givens, reasons, congruence, and similarity
After this book
College Algebra: Functions and Graphs
Domains, composition, inverses, and models