📘 Remember
- Label the triangle before choosing a formula.
- Use degree mode unless the question says otherwise.
- Keep full calculator values until the final rounding step.
🎯 Learning Objectives
- recognise when the sine rule applies;
- match sides with opposite angles;
- find missing sides and angles;
- handle the ambiguous case cautiously.
Introduction
This lesson develops The Sine Rule from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
The Sine Rule
a/sin A = b/sin B = c/sin C
When to Use It
Use the sine rule when you know an opposite side-angle pair and at least one other side or angle.
Worked Example
A=42°, a=9 cm and B=68°. Find b.
b/sin68°=9/sin42°
b=9sin68°/sin42°
b≈12.5 cm
Practice Questions
- A=35°, a=8 cm, B=74°. Find b.
- A=48°, a=12 cm, b=9 cm. Find B.
- What essential matching relationship must be identified first?
✅ Check Your Work
Show answers and working
- b=8sin74°/sin35°≈13.4 cm.
- sinB=9sin48°/12≈0.5574, so B≈33.9°. A second possible angle must be checked against the triangle sum and context.
- Each side must be paired with its opposite angle.
🧭 Interactive Lab: Sine Rule Pair Explorer
Change the values or make a choice and watch the mathematics respond.
Summary
⚠️ Common Mistakes
- Using the wrong reference angle.
- Rounding too early.
- Forgetting units or the degree symbol.
⭐ Exam Tips
- Write the formula before substituting.
- Sketch and label any diagram that is not already provided.
- Check whether the answer is sensible for the shape.
🌍 Did You Know?
Surveyors, engineers, pilots and architects use trigonometry to calculate distances that are difficult or unsafe to measure directly.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.