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
Share and support
Teaching Table
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
Full guide and materials
- Paper
- Pencil
- Ruler or measuring tape
- Calculator with inverse-trigonometric functions, used with adult guidance
- Step 1. Draw a right triangle and label one acute angle A. Mark the opposite side, adjacent side, and hypotenuse. Say SOH CAH TOA together.
- Step 2. For a 5 m ladder at 70° to the ground, identify the ladder as the hypotenuse and wall height as opposite. Use sin(70°) = height ÷ 5, so height ≈ 4.70 m.
- Step 3. For a triangle with opposite 6 and adjacent 8, use A = tan⁻¹(6 ÷ 8) ≈ 36.9°.
- Step 4. Explain why the selected ratio fits the labeled sides before using the calculator.
- Quick check. Give a new triangle with hypotenuse 10 cm and angle 35°, asking for the opposite side, then opposite 9 and adjacent 12 asking for the angle. Success means choosing sine and finding about 5.74 cm, then choosing inverse tangent and finding about 36.9°.
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.
Compare teaching approaches
Teaching approaches
Gakuva’s recommended starting approach is selected. You can still compare every option.
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
Keep a record of your learning
Save photos, notes, and attendance. Export a proof packet for compliance or portfolio reviews.
Learning resources
External links for extra practice. They do not replace Gakuva’s teaching approach.
No additional resources are listed for this topic.
The teaching approach appears above.
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