Sine, Cosine and Tangent · Hook

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Ever wondered how engineers know how steep to make a ramp?

A wheelchair ramp at Obi's school must not be too steep. Engineers link the angle of the ramp to its height and length with three special ratios: sine, cosine and tangent.

Revision notes

📒 Sine, Cosine and Tangent: lesson notes

Label the sides of a right-angled triangle and use SOH CAH TOA to write ratios and find missing sides.

A wheelchair ramp at Obi's school must not be too steep. Engineers link the angle of the ramp to its height and length with three special ratios: sine, cosine and tangent.

Naming the sides

In a right-angled triangle, pick one of the acute angles and call it θ (theta).

The hypotenuse is opposite the right angle (the longest side). The opposite side is across from θ — it doesn't touch θ. The adjacent side touches θ but is not the hypotenuse.

If you pick the other acute angle instead, the opposite and adjacent sides swap over; the hypotenuse stays the same.

🔑 Key idea: Hypotenuse first (opposite the right angle), then opposite and adjacent from θ.

SOH CAH TOA

Sin θ = Opposite ÷ Hypotenuse. Cos θ = Adjacent ÷ Hypotenuse. Tan θ = Opposite ÷ Adjacent.

If opposite = 3, adjacent = 4 and hypotenuse = 5, then sin θ = 35, cos θ = 45 and tan θ = 34.

These ratios depend only on the angle, not on the size of the triangle — similar triangles give the same ratios.

🔑 Key idea: SOH CAH TOA.

Finding a missing side

Choose the ratio that links the side you know and the side you want. Write it out, put in the numbers and solve.

If sin θ = 35 and the hypotenuse is 10 cm, then opposite ÷ 10 = 35, so opposite = 35 × 10 = 6 cm.

For angles like 35°, you read the ratio from tables or a calculator, for example tan 35° ≈ 0.7002.

🔑 Key idea: Write the ratio, substitute, then multiply (or divide) to find the side.

Worked example

In a right-angled triangle, for the angle θ: the opposite side is 12 cm, the adjacent side is 5 cm and the hypotenuse is 13 cm. Find tan θ as a fraction in its lowest terms.

  1. Step 1: Use TOA

    tan θ = opposite ÷ adjacent = 125

Answer: 125

Summary

  • The hypotenuse is opposite the right angle; opposite and adjacent depend on θ.
  • sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent.
  • Remember SOH CAH TOA.
  • To find a side: write the ratio, substitute and solve.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.