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Course 220 · Life course 42

Use Right-Triangle Trigonometry

A desk-based right-triangle investigation with labeled sides, ratio choices, checked distances and an uncertainty comparison. Six introductory lessons use supplied practice cases.

Before you start

Use paper or a free text document; every core input is supplied in the lessons.

Reason About Shapes with Geometry See everything you’ll need

Free download · Original practice data · No external account required. Download the starter pack

What you’ll leave with

A desk-based right-triangle investigation with labeled sides, ratio choices, checked distances and an uncertainty comparison.

Start course Start without an account. Sign in to save lesson progress across devices.

Your project

Start with the practice files. Build your project one lesson at a time.

Practice files are free without an account. Sign in to save your project record in My Proof: a description, checklist, and optional work link. Keep the original files in your own app or notebook; saving a project record does not upload them.

  1. Get your materialsDownload practice filesSupplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.
  2. Make your projectView the project checklist Work in your own notebook, document, or software. Use the supplied practice examples when you prefer.
  3. Keep a private recordSave a project record Sign in to save a description and checklist, then return to update them.
Course handbook

Know the route before you begin.

Here’s what you’ll need and what you’ll make. You can return to this guide during any lesson.

Who this is for

Learners aged 14+ building subject foundations and practical independence. Adults can use the same supplied cases.

Prepare before lesson one
  • Use paper or a free text document; every core input is supplied in the lessons.
  • A basic calculator is optional; show the method, units and checks.
  • Suggested earlier course: Reason About Shapes with Geometry
Your final project

Finish with proof you can use.

A desk-based right-triangle investigation with labeled sides, ratio choices, checked distances and an uncertainty comparison.

Save a project record in My Proof
See project checklist and review criteria

Keep your working files in your own notebook or software. Save a private description, checklist, and work link in My Proof.

What to include

  • Label all three sides relative to a chosen angle.
  • Check one Pythagorean calculation.
  • Choose ratios from known and unknown sides.
  • Show a distance and an inverse-angle calculation.
  • Include observer offset, uncertainty and all changed cases in the height model.

Review and revise

Use these criteria to check your work and choose what to improve.

  • Method: applies the course-specific rules to the supplied cases with visible intermediate work.
  • Accuracy: results agree with the stated facts, units, boundaries and independent checks.
  • Reasoning: explains why the result follows and identifies a limitation or unresolved question.
  • Transfer: completes changed cases and revises the final project when a relevant assumption changes.
Three ideas to help you along the way
Three useful ideas

The rules behind the steps.

01

Name the quantities before calculating

The hypotenuse lies opposite the right angle and is the longest side. Relative to one acute angle, opposite means across from it and adjacent means the touching leg; the hypotenuse is not the adjacent leg in a trigonometric ratio.

02

Use the correct relationship

For an acute angle in a right triangle, sin=opposite/hypotenuse, cos=adjacent/hypotenuse, tan=opposite/adjacent. Select the ratio connecting the known quantity and the requested quantity, not whichever button is convenient.

03

Keep the check and boundary visible

A right-triangle height model requires a horizontal distance and an angle above the horizontal. Add the observer height if the vertical estimate starts at eye level. Measurement uncertainty and model assumptions matter; no climbing or roadside fieldwork is required.

What do these words mean?
Key words

Key terms worth knowing.

Hypotenuse
Side opposite a right angle.
Opposite
Side across from the selected acute angle.
Adjacent
The leg next to the selected acute angle.
Tangent
Opposite leg divided by adjacent leg.
Inverse tangent
Operation finding an angle from its tangent ratio.
Model estimate
A result conditional on inputs and assumptions.
Your route

6 lessons.
One useful project.

Work in order the first time. Each lesson makes something the next lesson can use.

  1. 1
    Up next

    Identify right triangles and corresponding sides

    Label right-triangle sides relative to a selected acute angle.

  2. 2
    11 min

    Use the Pythagorean relationship

    Find an unknown right-triangle side using the Pythagorean relationship and a length check.

  3. 3
    11 min

    Choose sine, cosine or tangent

    Choose a trigonometric ratio from the known and requested right-triangle sides.

  4. 4
    13 min

    Calculate an unknown distance

    Calculate an unknown opposite distance from a supplied angle ratio and adjacent side.

  5. 5
    13 min

    Calculate an unknown angle

    Use an inverse trigonometric value to find an acute angle and its complement.

  6. 6
    15 min

    Check a height estimate and its limits

    Build a height estimate with observer offset and a stated input-uncertainty comparison.

Use the tools

Tools to help you practice.

See all tools
ZIP · practice files

Use Right-Triangle Trigonometry practice files

Supplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.

Text worksheet

Your course project worksheet

A downloadable text worksheet with each lesson’s working output, space for your answers, and the shared final-project checklist. Save your own copy.

Source transparency

Sources and further study.

Explore the references behind the lessons and take your learning further. Source notes explain access and reuse.

OpenStax: angles
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement. Exact topic chapters were opened in web result turn 325view 0–14.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: right-triangle trigonometry
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: unit circle
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

Ready when you are

Start with the first useful move.

You’ll always know what to do, what to make, and where to go next.

Start lesson one