eduKatePunggol · Secondary 3 Mathematics
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Match Opposite Pairs Before Using the Sine Rule. Begin with the explanation or choose the route closest to your question.
Route 1 · Chapters 1–2
Route 2 · Chapters 3–4
Route 3 · Chapters 5–6
Route 4 · Chapters 7–8
Full chapter index · Independent practice · Secondary 3 Mathematics support
If your child pairs a side with a nearby angle in the sine rule, mark the opposite side for each angle before entering any numbers. In triangle ABC, side BC is opposite angle A, side AC is opposite angle B, and side AB is opposite angle C. Secondary 3 Mathematics tuition in Punggol can make this matching step explicit.
For example, if A = 30°, B = 45° and BC = 10 cm, then AC/sin 45° = 10/sin 30°. Hence AC = 10√2 cm, approximately 14.1 cm. A Punggol Secondary 3 Maths tutor should ask why ten belongs with thirty degrees, not simply whether the calculator answer looks tidy.
This guide uses original idealised triangles and focuses on the side-angle pairing decision. It distinguishes the sine rule from right-triangle SOHCAHTOA and explains the extra check needed when finding an angle. Match the work to your child’s current school programme rather than assuming one topic sequence. Small-group 3-pax teaching should let every learner label the pairs independently.
Full chapter index
See the first decision · Chapters 1–2
Follow the worked method · Chapters 3–4
Check the conditions · Chapters 5–6
Practise and continue · Chapters 7–8
Chapter 1 of 8 · See the first decision
1. What does ‘opposite’ mean in triangle ABC?
The side opposite a vertex is the side not touching that vertex. Angle A is formed by AB and AC, so its opposite side is BC. Angle B is formed by BA and BC, so its opposite side is AC. This relationship comes from the labels, not the side’s visual position.
Trace the two sides meeting at an angle, then identify the remaining side. Rotating or reflecting the sketch does not alter the pairing. A side drawn at the bottom is not automatically side a. Standard notation uses a opposite A, but a question can use different letters, so read its definitions first.
For opposite pairs (a, A), (b, B) and (c, C), the sine rule is a/sin A = b/sin B = c/sin C. Its reciprocal form, sin A/a = sin B/b = sin C/c, is equally valid. Keep the orientation consistent across the equation.
A useful setup lists pairs before fractions: ‘BC, 30°’ and ‘AC, 45°’. Then form the two ratios. Pairing is more important than the order in which the ratios appear. Writing one ratio as side over sine and another as sine over side does not preserve the equality.
Chapter 3 of 8 · Follow the worked method
3. How does the 30°, 45°, 10 cm example work?
Given A = 30°, B = 45° and BC = 10 cm, the known complete pair is ten with thirty degrees. We want AC, opposite forty-five degrees. Write AC/sin 45° = 10/sin 30°, giving AC = 10 sin 45°/sin 30° = 10√2 cm.
The angle sum gives C = 105°, so the supplied angles form a valid triangle. The forty-five-degree angle exceeds thirty degrees, so AC should exceed BC. The result approximately 14.1 cm passes that ordering check. Keep the exact expression or full calculator precision until the requested final rounding.
Chapter 4 of 8 · Follow the worked method
4. How does a wrong pair change the result?
If a student writes AC/sin 30° = 10/sin 45°, the result becomes 5√2 cm, about 7.07 cm. That assigns the unknown side to the wrong angle. The algebra and arithmetic may be internally consistent, but the equation no longer describes the labelled triangle.
The smaller result also fails the side-ordering check: a larger angle should have the longer opposite side. This check is useful, but not sufficient by itself; some wrong setups can still produce plausible lengths. Return to the vertex labels and the complete opposite pair whenever the equation is in doubt.
SOHCAHTOA uses opposite, adjacent and hypotenuse within a right triangle relative to one chosen acute angle. The sine rule relates opposite side-angle pairs across any non-degenerate triangle, including non-right triangles. It does not require selecting a hypotenuse.
If a right angle is supplied, either an appropriate right-triangle ratio or the sine rule may be valid. Without a right angle, do not introduce a hypotenuse just because one side looks longest. Establish the type of triangle and the known information before choosing the relationship.
Rearranging may give a sine value, after which the calculator returns a principal inverse-sine angle. But sin θ = sin(180° − θ), so another angle can sometimes be possible. Check the triangle’s other angles and the supplied geometry before discarding or accepting a supplementary angle.
For instance, a sine value of 0.5 suggests 30° or 150° within a triangle. If another angle is already 50°, the 150° option is impossible because the sum would exceed 180°. If the other known angle were 20°, both options might remain possible under suitable side data. The diagram’s appearance alone is not a reliable decision rule.
Question 1: in triangle PQR, which side is opposite Q? Question 2: A = 30°, B = 90° and a = 5 cm; find b. Question 3: A = 30°, B = 60° and a = 6 cm; find b exactly. Question 4: sin B = 0.5 and A = 50°; which candidate for B can belong to a triangle?
Answers: 1) PR. 2) b = 5 sin 90°/sin 30° = 10 cm. 3) b = 6√3 cm, approximately 10.4 cm. 4) B = 30°; 150° is excluded by the angle sum. For Questions 2 and 3, ask for the two opposite pairs before the equation. That is the specific skill being checked.
Bring the original labelled diagram and the first equation your child wrote. A note such as ‘pairs AC with angle A because the letters look related’ gives the tutor a clear starting point. In a small-group 3-pax lesson, learners can label rotated triangles before solving numerical questions, then explain one pairing without help.
For parents considering Secondary 3 Mathematics tuition in Punggol, the next-question routine is: name the triangle, match opposite pairs, select a complete known pair, form a consistent equation, calculate and check side ordering or angle sum. The aim is to make the equation describe the triangle before asking the calculator to evaluate it.
