L2 · Sine, cosine, tangent: finding a side

Learning goal

“I can find an unknown side of a right triangle when I know one acute angle and one side: I choose the ratio that uses the known side and the unknown side, write the equation, and rearrange it — multiplying when the unknown is in the numerator and dividing when it is in the denominator.”

Reference angle — the marked acute angle the ratio is built from; opposite and adjacent are named relative to it.

Unknown (pronumeral) — the side we are solving for, written as a letter (x, h, …) inside the ratio equation.

Isolate — to rearrange an equation so the unknown stands alone on one side; whatever is done to one side is done to the other.

Worked example

A right triangle has hypotenuse 20 cm. The marked angle is 30°. Find the side opposite the 30° angle. Step 1: Sides in play: hypotenuse (known, 20) and opposite (unknown, x). The ratio that uses exactly those two is sin (SOH). Step 2: Write the definition with the angle and numbers: sin 30° = x/20. Step 3: x is in the numerator. Multiply both sides by 20: x = 20 × sin 30° = 20 × 0.5 = 10 cm. Sanity check: 10 < 20 ✓ (a leg must be shorter than the hypotenuse), and sin 30° = 0.5 exactly, so the opposite side is exactly half the hypotenuse — worth remembering.

Check yourself — explain it

In sin 30° = x/20, why do we multiply by 20 instead of dividing?

Paired problem — example still visible

Hypotenuse 15 cm, marked angle 40°, find the adjacent side -> 11.5 cm (cos 40° = x/15, so x = 15 × cos 40° = 15 × 0.766 = 11.5 cm)

Faded completion — last step is yours

"Marked angle 35°, adjacent side 9 m, opposite side x: tan 35° = x/9, so x = 9 × tan 35° = 9 × 0.700 = ___ m" -> 6.3 m

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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Grade 10 — Mathematics 10C › C10U03 Trigonometry of Right Triangles

L2 · Sine, cosine, tangent: finding a side