Ever wondered how a carpenter knows how long to cut a roof rafter?

Obi the carpenter knows the roof must slope at 30° and the rafter must rise 2 m. Without climbing up to measure, he can work out the exact length. The secret is that every right-angled triangle with a 30° angle has sides in the same ratio.

📒 Trigonometric Ratios: lesson notes

Use sine, cosine and tangent to find sides and angles in right-angled triangles, including the special angles 30°, 45° and 60°.

Naming the sides

In a right-angled triangle, the longest side (opposite the right angle) is the hypotenuse.

Pick one of the other angles, θ. The side across from θ is the opposite; the side next to θ (that is not the hypotenuse) is the adjacent.

The names depend on which angle you are looking from — always label from θ.

🔑 Key idea: Hypotenuse = longest side; opposite = across from θ; adjacent = next to θ.

SOH CAH TOA

sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.

These ratios stay the same for every triangle with the same angle θ, however big or small — that is why they are useful.

To find a side: write the ratio, put in what you know, and solve. If sin 40° = 0.6428 and the hypotenuse is 10 cm, the opposite side is 10 × 0.6428 = 6.428 ≈ 6.4 cm.

🔑 Key idea: SOH CAH TOA — choose the ratio that links the side you know and the side you want.

The special angles 30°, 45° and 60°

Cut an equilateral triangle of side 2 in half: you get sides 1, √3 and 2 with angles 30°, 60° and 90°. So sin 30° = 12, cos 30° = √3/2 and tan 60° = √3.

A right-angled isosceles triangle has sides 1, 1 and √2, so tan 45° = 1 and sin 45° = cos 45° = √2/2.

These give exact answers without tables: if the hypotenuse is 16 cm and the angle is 30°, the opposite side is 16 × 12 = 8 cm.

🔑 Key idea: sin 30° = cos 60° = ½, tan 45° = 1.

Worked example

In a right-angled triangle, tan θ = 724 and the adjacent side is 96 cm. Find the side opposite θ.

Step 1: Write the ratio
tan θ = opposite ÷ adjacent
724 = opposite ÷ 96
Step 2: Multiply by 96
opposite = 724 × 96 = 28 cm

Answer: 28 cm

Summary

Label the sides from the angle θ: opposite, adjacent, hypotenuse.
SOH CAH TOA: sin = O ÷ H, cos = A ÷ H, tan = O ÷ A.
To find a side, multiply (or divide) by the ratio value; to find an angle, work out the ratio first.
Special angles: sin 30° = cos 60° = ½, tan 45° = 1, sin 60° = cos 30° = √3/2.

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.