Trigonometry basics
Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem
Lesson
Set out your lesson
“In a right triangle, sine compares opposite to hypotenuse, cosine compares adjacent to hypotenuse, and tangent compares opposite to adjacent: SOH CAH TOA. The ratio must match the known and unknown sides. Inverse sine, cosine, or tangent finds an angle when the side lengths are known.”
Bring to the table
- Paper
- Pencil
- Ruler or measuring tape
- Calculator with inverse-trigonometric functions, used with adult guidance
Extra for the adult
Worked example. The ladder is the hypotenuse and the height is opposite the 70° ground angle, so sin(70°) = height/5. Therefore height = 5 × sin(70°) ≈ 4.70 m.
Optional challenge. Solve for the adjacent side when the hypotenuse is 12 cm and the angle is 40° using cosine, then explain why the result should be less than 12 cm.
What to notice. Save a photo of the labeled triangle and two equations, or a short parent note recording the chosen ratio, result, and explanation for one side and one inverse-angle calculation.
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Teaching approaches
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1 approach
In a right triangle, sine compares opposite to hypotenuse, cosine compares adjacent to hypotenuse, and tangent compares opposite to adjacent: SOH CAH TOA. The ratio must match the known and unknown sides. Inverse sine, cosine, or tangent finds an angle when the side lengths are known.
Spend 15 minutes labeling a right triangle, choosing the correct ratio, and solving one side problem and one angle problem with a calculator in degree mode.
Family records and proof options
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Save photos, notes, and attendance. Export a proof packet for compliance or portfolio reviews.
Learning resources
Extra practice links — they do not replace Gakuva’s teaching approach.
PhET Interactive Simulations · Practice link — not the full lesson
Area Builder
Before you open · source notes
Area Builder supports hands-on exploration for Trigonometry basics. Not every lab worksheet.
Free interactive page, no sign-in required for the tested path.
Before you openAsk the learner what they changed in the sim or page and what they observed.
Opening this link does not save learning or assess mastery.
Source, access and review notes
Public PhET Interactive Simulations page. An anonymous GET on 2026-11-12 returned HTTP 200 with no login or payment prompt for this URL. Opening the link does not save practice or assess mastery. An adult should preview and guide the learner.
PhET Interactive Simulations. “Area Builder.” External page; Gakuva does not copy, host, embed, or redistribute materials.
Unknown — link only
Check understanding
- Identify which trigonometric ratio to use (SOH CAH TOA) based on the known and unknown sides
- Calculate an unknown side using sin, cos, or tan and a known angle
- Find an unknown angle using the inverse trigonometric function (e.g. tan⁻¹)
“If your child knows a ladder is 5 metres long and leans against a wall at 70° to the ground, can they use trigonometry to work out how high up the wall it reaches?”
Curriculum record
- Type
- Procedural
- Subject
- Mathematics
- Domain
- Geometry
- Age range
- Ages 13–15
Standards