Grade 10 · Grade 10: Circle Geometry and Trigonometry · Circle Geometry, Trigonometry and Solids
Sine Rule, Cosine Rule and Solids
Solve any triangle, then apply the same reasoning to surface area and volume of composite solids.
Learning objectives
- Apply the sine rule and the cosine rule.
- Use Area = ½ab·sin C.
- Solve problems involving surface area and volume of prisms, pyramids, cylinders, cones, spheres and composite solids.
- Work with right triangles inside three-dimensional figures.
AERO Mathematics alignment
AERO.M10.GM.3
Apply the sine rule, the cosine rule and the formula Area = ½ab·sin C to find unknown measurements in any triangle.
AERO.M10.GM.5
Solve problems involving the surface area and volume of prisms, pyramids, cylinders, cones, spheres and composite solids, including right-triangle work in three dimensions.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A surveyor knows two sides of a triangular plot and the angle between them, but the plot has no right angle.
Think about it
Can SOH-CAH-TOA still help, and if not, what will?
Hint: You need a rule that works for every triangle.
EXPLAIN
Solving any triangle
| Rule | Formula | Use when you know |
|---|---|---|
| Sine rule | a/sin A = b/sin B = c/sin C | Two angles and a side, or two sides and a non-included angle |
| Cosine rule | c² = a² + b² − 2ab·cos C | Two sides and the included angle, or all three sides |
| Area | Area = ½ab·sin C | Two sides and the included angle |
For the surveyor with a = 60 m, b = 80 m and C = 55°, the cosine rule gives c² = 3600 + 6400 − 2(60)(80)cos 55° ≈ 4494, so c ≈ 67 m, and the area is ½(60)(80)sin 55° ≈ 1966 m².
The ambiguous case
Because sin θ = sin(180° − θ), the sine rule can give two valid angles when you know two sides and a non-included angle. Always check whether both possibilities fit a triangle whose angles sum to 180°.
Solids and composite figures
Break composite solids into prisms, pyramids, cylinders, cones and spheres, and add or subtract. Right triangles inside a solid, such as the slant height of a cone, connect trigonometry to three-dimensional measurement.
Remember
Three sides means the cosine rule. Matching angle-and-opposite-side pairs means the sine rule.
INVESTIGATION
Try it yourself
Map a real triangular space by measuring two sides and the included angle. Calculate the third side and the area, then verify the third side with a tape measure and report the percent error.
Watch
Law of Cosines
The Organic Chemistry Tutor · 10:17
Shows when to reach for the cosine rule instead of the sine rule.
Before you watch: Which rule do you use when you know three sides?
- Find the third side given a = 7, b = 9 and C = 52°.
- Find the area of a triangle with a = 6, b = 8 and C = 40°.
Interactive simulation · PhET
Trig Tour
Explore sine values for obtuse angles to understand the ambiguous case.
While you explore
- Why can two different angles share the same sine value?
- When should you use the cosine rule instead of the sine rule?
Key vocabulary
- Ambiguous case
- When the sine rule gives two possible triangles.
- Cosine rule
- A formula linking three sides and one angle in any triangle.
Practice questions
0/1 correct
Level 1 · Criterion A
You know all three sides of a triangle and want an angle. Which rule should you use?
MYP criterion tasks
Level 2 · Criterion D
Find the area of a triangle with sides 9 cm and 14 cm and an included angle of 40°.
Reflect & track
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