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MathidiaMathematics Academy
Non-right-angled trigonometry
Lesson 36 of 75
Introduction

Sine rule for sides

Objective. Calculate missing sides in non-right-angled triangles.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The sine rule for sides lets you find a missing side length in any triangle, not just right-angled ones, when you know an angle and its opposite side plus one other angle. The rule states a/sin A = b/sin B = c/sin C, so rearranging gives the unknown side as (known side × sin of its opposite angle) divided by sin of the other known angle. For a triangle with angle A = 40°, angle B = 60° and side a = 8 cm, side b = 8 × sin60°/sin40° ≈ 10.78 cm.

Worked example

In triangle ABC, angle A = 50°, angle C = 70°, and side a = 12 cm. Use the sine rule for sides to find side c.

Diagram of a triangle
  1. 1Write the sine rule: a/sin A = c/sin C
  2. 2Substitute known values: 12/sin50° = c/sin70°
  3. 3Rearrange: c = 12 × sin70°/sin50°
  4. 4Calculate: c = 12 × 0.9397/0.7660 ≈ 14.72 cm

Answer: c ≈ 14.72 cm

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Common mistake. Students sometimes pair a side with the wrong angle, dividing by the angle that is not actually opposite that side. Always check carefully which angle is directly opposite each side before setting up the ratio.

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