The Sine Rule | Better Skills Academy Student Hub

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Watch a worked example

The sine rule links every side of a triangle with the angle directly opposite it. Follow each step below — click through at your own pace.

Example 1 — find a missing side

Triangle ABC: ∠A = 48°, ∠B = 72°, side a = 9.5 cm (opposite A). Find side b.

Write the sine rule, matching each side with the angle opposite it: asin A = bsin B

Substitute the known values: 9.5sin 48° = bsin 72°

Make b the subject by cross-multiplying: b = 9.5 × sin 72°sin 48°

Evaluate on your calculator (degree mode): b ≈ 12.2 cm (1 d.p.)

Example 2 — find a missing angle

Triangle PQR: ∠P = 100°, side p = 14, side q = 9.8. Find ∠Q.

The sine rule works just as well upside-down, which is easier when solving for an angle: sin Qq = sin Pp

Substitute the known values: sin Q9.8 = sin 100°14

Make sin Q the subject: sin Q = 9.8 × sin 100°14 ≈ 0.689

Take sin⁻¹ of both sides: ∠Q ≈ 43.6°

Watch out — the ambiguous case. sin⁻¹ can give two possible angles between 0° and 180°, since sin(180° − θ) = sin(θ). In Example 2, ∠P was already obtuse (100°), so ∠Q had to be acute and there was only one valid answer. You'll meet a problem with genuinely two valid answers in Question 5 of today's practice.

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Your turn, with help

Work through this problem yourself — fill in each blank, then click Check. You can try as many times as you like.

Guided example

Triangle XYZ: ∠X = 65°, ∠Y = 55°, side x = 18 m (opposite X). Find side y.

Step 1. Write the sine rule linking the side we know with the side we want, each matched to the angle opposite it: sin X = ysin

Step 2. Substitute the known values: 18sin 65° = ysin °

Step 3. Rearrange to make y the subject and evaluate: y = (18 × sin 55°) / sin 65° ≈ m (1 d.p.)

Show full solution

x/sin X = y/sin Y → 18/sin 65° = y/sin 55° → y = (18 × sin 55°)/sin 65° ≈ 16.3 m

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Practice on your own

Five questions, from straightforward to a genuine challenge. Type your answer and click Check — if you'd rather just see how it's done, every question has a full worked solution underneath.

Question 1

Triangle ABC: ∠A = 50°, ∠B = 65°, side a = 8 cm. Find side b.

b ≈ cm

Show full solution

b/sin B = a/sin A → b = (8 × sin 65°)/sin 50° ≈ 9.46 cm

Question 2

Triangle with ∠C = 80°, ∠A = 35°, side c = 15 m. Find side a.

a ≈ m

Show full solution

a/sin A = c/sin C → a = (15 × sin 35°)/sin 80° ≈ 8.74 m

Question 3

Triangle with ∠P = 110°, side p = 22, side q = 14. Find ∠Q.

∠Q ≈ °

Show full solution

sin Q/q = sin P/p → sin Q = (14 × sin 110°)/22 ≈ 0.598 → ∠Q ≈ 36.7° (acute only, since ∠P is already obtuse)

Question 4

Triangle with ∠A = 28°, ∠B = 64°, side b = 10. Find side c. (Hint: find ∠C first.)

c ≈

Show full solution

∠C = 180° − 28° − 64° = 88°. Then c/sin C = b/sin B → c = (10 × sin 88°)/sin 64° ≈ 11.1

Question 5Challenge — two answers

Triangle with ∠B = 30°, side b = 5, side c = 9. Find ∠C. Enter either valid answer.

∠C ≈ °

Show full solution

sin C = (9 × sin 30°)/5 = 0.9. sin⁻¹(0.9) gives ∠C ≈ 64.2° OR 180° − 64.2° = 115.8°. Both are valid here because ∠B = 30° is small enough that either value of ∠C still leaves a positive ∠A (180° − B − C > 0°) — this is the ambiguous case flagged in the tip above.