IGCSE Maths / Further trigonometry

The sine rule: finding a missing angle

Edexcel International GCSE Maths (4MA1), chapter 25: Further trigonometry, specification point 4.8c. Free lesson.

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Use the sine rule with the sines on top when you need to find a missing angle. Match each side to the angle opposite it, then use inverse sine to turn the sine value into an angle.

Write the sine rule with sines on top

The formula sheet gives the sine rule as side divided by sine. When an angle is missing, turn both fractions upside down so the sines are on top. The fractions stay equal because flipping both sides of an equation preserves equality. Sine rule for finding an angle: (sin A)/a = (sin B)/b Each side is paired with the angle opposite it.

Match the known opposite pair

In triangle ABC, angle A is 40°, BC is 9 cm and AC is 7 cm. BC faces A, so 40° and 9 cm are the known pair. AC faces B, so 7 cm belongs with the angle you are finding. Use (sin B)/7 = (sin 40°)/9. Rearrange to get sin B = (7 × sin 40°)/9 ≈ 0.4999.

Turn the sine value into an angle

The result of the rearrangement is a sine value, not the angle itself. Use the inverse sine key, sin⁻¹, to work backwards from that value to the angle. The same method works when the letters change. For angle C, AC = 12.4 cm faces the known angle B = 62°. AB = 9.6 cm faces C, so sin C = (9.6 × sin 62°)/12.4. Using inverse sine gives C ≈ 43.1°.

Check the angle against the triangle

Check whether your answer fits the side lengths in the diagram. A shorter side must face a smaller angle. This can reveal a swapped pair even when the answer looks reasonable. In the first triangle, B faces the 7 cm side, so it must be smaller than the 40° angle facing 9 cm. If the 7 and 9 are swapped, sin B = (9 × sin 40°)/7 ≈ 0.826, giving about 55.7°. This does not fit the triangle.

Use the same method for any letters

For triangle PQR, QR = 10.5 cm faces P = 48°, and PR = 8.2 cm faces Q. Put those opposite pairs into the sine rule, then use inverse sine and check that the answer fits the side lengths. Start with (sin Q)/8.2 = (sin 48°)/10.5. The answer is Q ≈ 35.5° to 3 significant figures.

More lessons in Further trigonometry

• The sine rule: finding a missing side
• The ambiguous case of the sine rule
• The cosine rule: finding a missing side
• The cosine rule: finding a missing angle
• Choosing between the sine rule and the cosine rule
• Area of a triangle using half ab sin C
• Finding a side or an angle from a given area
• Area of a segment
• The angle between a line and a plane
• Trigonometry in pyramids and cones
• Problem solving in three dimensions