Trigonometry basics

MathematicsGeometryAges 13–15

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

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“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.”

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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.

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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.

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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

uk-nc-2013:KS3.Maths.Geo.14