📘 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 cosine rule applies;
- find a side from two sides and included angle;
- find an angle from three sides;
- label sides opposite their matching angles.
Introduction
This lesson develops The Cosine 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 Cosine Rule
a²=b²+c²−2bc cos A
Use it for SAS (two sides and included angle) or SSS (three sides).
Worked Example: Find a Side
b=7 cm, c=10 cm and A=60°.
a²=7²+10²−2(7)(10)cos60°
a²=49+100−70=79
a=√79≈8.89 cm
Worked Example: Find an Angle
a=8, b=6, c=5.
cosA=(b²+c²−a²)/(2bc)
cosA=(36+25−64)/(60)=−0.05
A=cos⁻¹(−0.05)
A≈92.9°
Practice Questions
- b=8, c=11, A=50°. Find a.
- a=9, b=7, c=6. Find A.
- Why can cosine be negative when finding an obtuse angle?
✅ Check Your Work
Show answers and working
- a²=64+121−176cos50°≈71.87, so a≈8.48.
- cosA=(49+36−81)/(84)=4/84, so A≈87.3°.
- Cosine values are negative for angles between 90° and 180°.
🧭 Interactive Lab: Cosine Rule 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.