Secondary Mathematics: Sine Rule, Cosine Rule and Non-Right Triangles

Secondary Mathematics · Worked Repair Guide 34

Right-triangle trigonometry works only when a 90° angle is available. In a general triangle, the sine rule and cosine rule extend trigonometric reasoning to situations where no right angle is given. The difficulty is not remembering two formulas; it is deciding which relationship fits the information already known.

This guide develops one central habit: pair every side with the angle opposite it before choosing a rule. The sine rule works through opposite side-angle pairs. The cosine rule connects three sides with the included angle. The area formula 1/2 ab sin C uses two sides and their included angle.

All triangles, bearings and distances below are original teaching constructions. Use Trigonometry, Pythagoras, Bearings and Elevation first when right-triangle sine, cosine, tangent or bearing conventions are unstable.

1. Standard triangle notation links sides to opposite angles

In triangle ABC, side a lies opposite angle A, side b opposite B and side c opposite C.

This pairing is structural. The side named by a lowercase letter is opposite the matching uppercase angle.

Entry check: if angle B is 70°, the side paired with it in the sine rule is b, not one of the sides touching angle B.

Mark the opposite pairs on the diagram before writing a formula.

2. The sine rule connects opposite side-angle pairs

The sine rule can be written a/sin A=b/sin B=c/sin C.

Equivalently, sin A/a=sin B/b=sin C/c.

Either orientation is valid; choose the one that keeps the unknown convenient.

The rule becomes useful when at least one complete opposite pair is known.

3. Use the sine rule to find a side

Suppose A=40°, B=65° and a=8 cm. Find b.

b/sin65°=8/sin40°.

Therefore b=8 sin65°/sin40°≈11.28 cm.

Check the ordering: B is larger than A, so side b should be longer than side a. The numerical result agrees.

4. Use the sine rule to find an angle

Suppose a=10 cm, b=7 cm and A=50°. Find B.

sin B/7=sin50°/10, so sin B=0.7 sin50°≈0.5362.

The principal acute solution is B≈32.4°.

Because sine can have the same positive value for two supplementary angles, a second candidate 180°−32.4°=147.6° should be checked against the triangle angle sum and side ordering where the ambiguous case is in scope.

5. The ambiguous sine-rule case needs a triangle check

For positive sine value sin B=k with 0<k<1, two angles between 0° and180° can share that sine: B and180°−B.

Not both candidates necessarily fit the triangle. They must leave a positive third angle and agree with side-size constraints.

In the previous example, A=50° and alternative B=147.6° would already make A+B=197.6°>180°, impossible. So only B≈32.4° survives.

Do not automatically accept only the inverse-sine calculator output when the course expects ambiguous-case reasoning.

6. The cosine rule extends Pythagoras

The cosine rule can be written c²=a²+b²−2ab cos C.

The angle C is the included angle between sides a and b, and side c lies opposite it.

If C=90°, cos90°=0, so the formula becomes c²=a²+b²: Pythagoras.

The cosine rule is therefore a generalisation rather than an unrelated theorem.

7. Use the cosine rule for two sides and the included angle

Suppose a=7 cm, b=10 cm and C=60°.

c²=7²+10²−2(7)(10)cos60°=49+100−70=79.

Thus c=√79≈8.89 cm.

The positive root is used because c is a length.

8. Use the cosine rule to find an angle from three sides

Suppose a=5, b=7, c=8. Find angle C.

8²=5²+7²−2(5)(7)cos C.

64=74−70 cos C, so cos C=10/70=1/7.

C=cos⁻¹(1/7)≈81.8°.

Since c=8 is the longest side, C should be the largest angle. That gives a useful ordering check.

9. Rearranging cosine-rule angle questions protects signs

From c²=a²+b²−2ab cos C, isolate cosine:

2ab cos C=a²+b²−c².

Therefore cos C=(a²+b²−c²)/(2ab).

Writing this form before substituting reduces sign errors caused by moving the negative cosine term mentally.

10. Triangle area can use two sides and the included angle

The area of a triangle can be written Area=1/2 ab sin C when sides a and b include angle C.

For a=8 cm, b=11 cm and C=35°, area=1/2(8)(11)sin35°≈25.24 cm².

The angle must be the included angle between the two sides used in the product.

Do not select any two side lengths and any convenient angle from the triangle.

11. Derive the sine-area formula from a perpendicular height

Take sides a and b with included angle C. Drop a perpendicular from the endpoint of side a to side b.

The height is a sin C.

Then area=1/2×base b×height a sin C=1/2 ab sin C.

The formula therefore comes directly from ordinary triangle area plus right-triangle sine.

12. Choose sine rule when a complete opposite pair is available

Suppose A=42°, a=9 cm and B=71°. The pair (A,a) is complete, so the sine rule directly connects to side b.

Using the cosine rule would require information not yet known.

A strong method choice uses the given structure rather than a favourite formula.

Mark the known opposite pair visually before calculating.

13. Choose cosine rule for SAS or SSS structure

Two sides and the included angle (SAS) naturally lead to the cosine rule for the third side.

Three sides (SSS) naturally lead to the cosine rule for an angle.

The sine rule cannot start from three sides alone because no angle-side pair is known.

Once one angle has been found, the sine rule may become a useful second-stage method.

14. Triangle inequalities provide a feasibility check

Three positive lengths form a non-degenerate triangle only when each is less than the sum of the other two.

Lengths 3,4,8 cannot form a triangle because3+4<8.

If a cosine-rule calculation appears to use impossible side data, the issue may be the given model rather than arithmetic.

Check feasibility before searching for an angle.

15. The largest side is opposite the largest angle

In any non-degenerate triangle, side order and opposite-angle order agree.

If a>b, then A>B.

This provides a quick reasonableness check for sine-rule and cosine-rule results.

An answer claiming the smallest angle lies opposite the largest side should trigger review.

16. Bearings often create non-right triangles

Suppose B is 8 km from A on bearing040°, and C is 11 km from A on bearing120°.

The included angle BAC is120°−40°=80°.

Cosine rule gives BC²=8²+11²−2(8)(11)cos80°.

BC≈11.08 km.

The bearing conventions create the triangle angle; the cosine rule then handles its non-right geometry.

17. Multi-step problems may change method after one result

Suppose a=6, b=9 and C=50°. First use cosine rule to find c.

c²=36+81−108 cos50°≈47.58, so c≈6.90.

Now a complete opposite pair (c,C) is known, so sine rule can find A:

sin A/6=sin50°/6.90, giving A≈41.8°.

The method changed because the information state changed.

18. Keep calculator precision through chained calculations

If a side found by cosine rule is immediately used in a sine-rule calculation, keep the full calculator value rather than rounding to two decimal places first.

Premature rounding can shift a later angle or length.

Record a sensible displayed value for communication while using stored precision for subsequent calculations.

Round the final answer according to the question’s instruction.

19. Capstone: area, side and angle in one triangle

An invented triangular plot has sides AB=12 m, AC=9 m and included angle A=55°.

Area=1/2(12)(9)sin55°≈44.23 m².

Side BC²=12²+9²−2(12)(9)cos55°≈101.10, so BC≈10.05 m.

Now sine rule gives sin B/9=sin55°/10.05, so B≈47.2°.

Then C≈77.8°. The largest angle C lies opposite the longest side AB=12, providing an independent ordering check.

20. Independent practice

1. In triangle ABC, which side is opposite angle B?
2. A=40°,B=65°,a=8. Find b.
3. A=50°,a=10,b=7. Find the principal value of B.
4. Explain why a second sine-rule angle candidate may need checking.
5. a=7,b=10,C=60°. Find c.
6. a=5,b=7,c=8. Find C.
7. Rearrange c²=a²+b²−2ab cos C to make cos C the subject.
8. Find the area for a=8,b=11,C=35°.
9. A=42°,a=9,B=71°. Choose the most direct rule for b and explain.
10. Sides 6 and9 include angle50°. Choose the most direct rule for the third side.
11. Do lengths3,4,8 form a triangle?
12. Do lengths5,7,8 form a triangle?
13. If one side is longest, what can be said about its opposite angle?
14. Two rays from A have lengths8 km and11 km and bearings040° and120°. Find their included angle.
15. Using Question14, estimate the distance between the endpoints.
16. For sides6,9 with included angle50°, find the third side.
17. Using the result of Question16, find the angle opposite side6.
18. Find the area of a triangle with sides12 and9 including angle55°.
19. Explain why 1/2 ab sin C requires C to be the included angle between a and b.
20. Explain how the cosine rule reduces to Pythagoras when C=90°.

21. Worked answers

1. Side b.

2. Approximately11.28. b=8sin65°/sin40°.

3. Approximately32.4°.

4. Because sin θ has the same positive value for supplementary angles; the alternative must be tested against angle sum and triangle conditions.

5. √79≈8.89.

6. Approximately81.8°.

7. cos C=(a²+b²−c²)/(2ab).

8. Approximately25.24 square units.

9. Sine rule. A complete opposite pair (A,a) is already known.

10. Cosine rule. Two sides and their included angle are known.

11. No. 3+4<8.

12. Yes. Each side is shorter than the sum of the other two.

13. Its opposite angle is the largest.

14. 80°.

15. Approximately11.08 km.

16. Approximately6.90.

17. Approximately41.8°.

18. Approximately44.23 square units.

19. The perpendicular height relative to base b is a sin C only when C is the angle between the two chosen sides.

20. cos90°=0, so c²=a²+b²−2ab(0)=a²+b².

22. Diagnose non-right-triangle errors by pairing first

Common failures include matching a side with an adjacent rather than opposite angle, using sine rule without a complete pair, using cosine rule with a non-included angle in SAS data, forgetting the ambiguous sine case, or rounding an intermediate side too early.

A useful correction note says “side a is opposite A”, “SAS/SSS → cosine rule”, “opposite pair → sine rule”, or “check supplementary sine candidate”.

Then rotate or relabel the triangle. A secure learner should reconstruct the pairings rather than depend on the side being drawn in a familiar position.

23. Continue through the BTT learning routes

Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Trigonometry, Pythagoras, Bearings and Elevation for right-triangle foundations and Angles, Similarity and Geometric Reasoning for general geometry.

Within Batch09, continue to Matrices, Data, Scalar Operations and Matrix Multiplication, Power Functions, Exponential Graphs and Tangent Gradients, or Statistical Diagrams, Stem-and-Leaf, Pie Charts and Misleading Displays.

24. Sources and scope

The triangles, bearings and practice problems are original teaching material. Sine rule, cosine rule and the sine-area relationship are used in their standard Euclidean-triangle sense.

For current Singapore Secondary curriculum documents and assessment scope, consult the relevant MOE/SEAB syllabus for the learner’s subject level and year. Match ambiguous-case depth and bearing synthesis to the actual course being studied.