Sine Rule, Knowledge Organiser

Key Terms

• Sine rule: Connects any side with its OPPOSITE angle
• Matched pair: A side and its directly opposite angle (e.g. side a and angle A)
• Non-right-angled triangle: A triangle with no 90° angle
• Included angle: The angle between two known sides (use the cosine rule for these)

Must-Know Facts

• Use the sine rule when you have a MATCHED PAIR plus one more piece of information
• Side a is ALWAYS opposite angle A (same letter)
• To find a side: use a/sin A = b/sin B
• To find an angle: flip to sin A/a = sin B/b, then use sin⁻¹
• The three angles of any triangle sum to 180°

Key Formulas

• Find side: a/sin A = b/sin B = c/sin C
• Find angle: sin A/a = sin B/b = sin C/c
• Use when: AAS, ASA (have angle + opposite side)

Common Mistakes

• Wrong pairing: Each side must pair with the angle directly opposite it: a pairs with A, b with B, c with C
• Finding an angle: Use sin A / a = sin B / b, then sin⁻¹ on your calculator, and check the three angles add up to 180°
• Using sine rule when cosine rule needed: Sine rule needs an angle-opposite-side pair: use cosine rule for SAS or SSS
• Calculator mode: Must be in degrees mode for GCSE problems

Worked Examples

In triangle ABC, A = 40°, B = 75°, and a = 8cm. Find b. Solution a/sin A = b/sin B 8/sin 40° = b/sin 75° b = 8 × sin 75° / sin 40° b = 8 × 0.966 / 0.643 = 12.0 cm (1 d.p.)