eduKatePunggol · Secondary 3 Mathematics
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Cosine Rule: Use the Angle Between the Known Sides. 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 inserts a convenient angle into the cosine rule, identify which two sides meet at that angle. In c² = a² + b² − 2ab cos C, C is the included angle between sides a and b, while c is opposite C. Secondary 3 Mathematics tuition in Punggol can make that relationship visible before substitution.
For two sides 5 cm and 8 cm enclosing 60°, the opposite side satisfies c² = 25 + 64 − 80 cos 60° = 49, giving c = 7 cm. A Punggol Secondary 3 Maths tutor should ask why sixty degrees belongs with those two lengths, not simply whether the calculator produces a positive answer.
This guide uses original idealised triangles and distinguishes the side-finding and angle-finding forms of the rule. It does not assume a universal school topic sequence. Small-group 3-pax support should help every learner identify the included angle, opposite side and stated geometry independently.
Full chapter index
Match the sides and angle · Chapters 1–2
Follow the worked calculation · Chapters 3–4
Check other triangle cases · Chapters 5–6
Practise and continue · Chapters 7–8
The included angle is formed by the two selected sides at their shared vertex. In triangle ABC, the sides AB and AC meet at A, so the angle between them is A. The remaining side BC is opposite A. Draw or point to those two arms before deciding which angle appears in the formula.
An angle touching just one of the selected sides is not their included angle. A label’s closeness to a printed number does not establish a relationship. Use the vertex and side endpoints rather than the page layout. Rotating the triangle does not change the matching.
Chapter 2 of 8 · Match the sides and angle
2. How should the formula match the diagram?
With a opposite A, b opposite B and c opposite C, the rule is c² = a² + b² − 2ab cos C. The two sides inside the product meet at C. Equivalent versions can be written for A or B, provided the opposite side is changed consistently.
For sides AB and AC known, a convenient form is BC² = AB² + AC² − 2(AB)(AC) cos A. Writing the actual side names can be clearer than forcing every problem into remembered letters. The geometry stays the same whichever notation is chosen.
Chapter 3 of 8 · Follow the worked calculation
3. How does the 5 cm, 8 cm, 60° example work?
Square the known sides to get 25 and 64. Their product term is 2 × 5 × 8 = 80, and cos 60° = 1/2. Thus c² = 89 − 40 = 49. Since c is a positive length, c = 7 cm. Keep the final square-root step separate from the squared-length result.
The triangle inequality gives a quick check: the third side must be greater than |8 − 5| = 3 and less than 8 + 5 = 13. Seven fits. This check can reject impossible values, but it does not prove the angle was chosen correctly because some wrong substitutions also produce lengths between three and thirteen.
Chapter 4 of 8 · Follow the worked calculation
4. Why can the wrong angle still give a plausible answer?
If thirty degrees were incorrectly used between those same known lengths, the calculation would give c² = 89 − 40√3, a positive number. Its square root is approximately 4.44 cm. That lies between three and thirteen, so a simple triangle-inequality check would not catch the mistaken angle.
The equation describes a different triangle: one with an included angle of thirty degrees, not the stated sixty. The repair is to return to the angle’s two arms. A calculator evaluates the equation it receives; it cannot determine whether that equation matches the labelled question.
Rearrange the same rule to find an angle: cos C = (a² + b² − c²)/(2ab). The side subtracted in the numerator is the side opposite the angle being found. For a = 5, b = 8 and c = 7, cos C = (25 + 64 − 49)/80 = 1/2, giving C = 60°.
Use inverse cosine, not division by cosine, to recover the angle. For a valid non-degenerate triangle, the computed cosine lies strictly between −1 and 1. A value outside that range can indicate inconsistent side data, a copying error or an incorrect setup. Do not force an impossible calculator entry into a triangle answer.
Chapter 6 of 8 · Check other triangle cases
6. How does the rule connect to right and obtuse triangles?
If C = 90°, cos C = 0 and the rule becomes c² = a² + b². This recovers Pythagoras for the side opposite the right angle. It does not justify applying Pythagoras when no right angle is supplied or established.
If C is obtuse, cos C is negative, so subtracting 2ab cos C adds a positive contribution. The opposite side is then longer than it would be with the same a and b enclosing ninety degrees. For a = b = 4 and C = 120°, c² = 16 + 16 + 16 = 48, so c = 4√3 cm.
Question 1: in triangle PQR, which angle is included between PQ and PR? Question 2: sides 3 cm and 4 cm enclose 90°; find the opposite side. Question 3: sides 4 cm and 4 cm enclose 60°; find the opposite side. Question 4: for sides a = 5, b = 8, c = 7, write the expression for cos C.
Answers: 1) Angle QPR, at P. 2) Five centimetres. 3) c² = 16 + 16 − 32 × 1/2 = 16, so four centimetres. 4) (25 + 64 − 49)/(2 × 5 × 8) = 1/2. Ask your child to identify the opposite side before calculating.
Chapter 8 of 8 · Practise and continue
8. What should parents and tutors check next?
Bring the original labelled diagram and the first formula line. A precise note is ‘uses an angle not enclosed by the two known sides’. In a small-group 3-pax lesson, students can annotate several differently oriented triangles and explain their selections before any calculator work.
For parents considering Secondary 3 Mathematics tuition in Punggol, the next-question routine is to match two sides, their included angle and the opposite side, then calculate and check. If the geometry is correct but the number differs, inspect degree mode, brackets, signs and the final square root separately.
