The idea, step by step
The cosine rule extends Pythagoras
The side opposite the included angle is found from the other two sides. The correction term accounts for the angle between those sides.
At 90° the cosine term is zero and the rule becomes Pythagoras. For an obtuse included angle its cosine is negative, so subtracting the cosine term increases the opposite side’s square.
Rearrange to find an angle
With all three side lengths known, isolate cosC before applying inverse cosine. The largest side must face the largest angle, providing an independent check.
First check triangle inequality: the two shorter sides must add to more than the longest. An invalid cosine result may reveal impossible lengths rather than a calculator problem.
A common trap
Starting with sine rule when no opposite side-angle pair is known.
Instead: For SAS or SSS information, cosine rule supplies the missing information.
See the thinking
Worked example 1
Two sides are 7 cm and 10 cm and their included angle is 60°. Find the opposite side.
- c²=7²+10²−2×7×10cos60°.
- c²=49+100−70=79.
- Take the positive square root.
Worked example 2
A triangle has sides 3 cm, 5 cm and 7 cm. Find its largest angle.
- The largest angle C is opposite 7 cm.
- cosC=(3²+5²−7²)/(2×3×5)=−0.5.
- C=cos⁻¹(−0.5)=120°.
Check your understanding
Find the angle opposite 10 in a triangle with sides 6, 8 and 10.
Show answer
90°
cosC=(36+64−100)/96=0.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Two sides are 8 m and 11 m and their included angle is 50°. Find the third side to one decimal place.2 marks
- Find the angle opposite 13 in a triangle with sides 9, 10 and 13. Round to one decimal place.2 marks
- Sides of 12 cm and 17 cm enclose 40°. Find the opposite side to one decimal place.2 marks
- Use cosine rule to show that sides 8, 15 and 17 form a right triangle.3 marks
- Could sides 4 cm, 5 cm and 12 cm form a triangle? Explain before calculating an angle.3 marks
- For a=6, b=9, C=110°, a student adds 2abcosC instead of subtracting it. Correct the rule and find c.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Two sides are 8 m and 11 m and their included angle is 50°. Find the third side to one decimal place.2 marks
c²=8²+11²−2×8×11cos50°.
Take the positive square root.
8.5 m - Find the angle opposite 13 in a triangle with sides 9, 10 and 13. Round to one decimal place.2 marks
cosC=(81+100−169)/(2×9×10).
C=cos⁻¹(12/180).
86.2° - Sides of 12 cm and 17 cm enclose 40°. Find the opposite side to one decimal place.2 marks
c²=144+289−408cos40°.
Take the positive square root.
11.0 cm - Use cosine rule to show that sides 8, 15 and 17 form a right triangle.3 marks
cosC=(64+225−289)/(2×8×15)=0.
Thus C=90°.
The angle opposite 17 is 90°. - Could sides 4 cm, 5 cm and 12 cm form a triangle? Explain before calculating an angle.3 marks
4+5=9<12.
The two shorter sides cannot close a triangle around the longest side.
No. - For a=6, b=9, C=110°, a student adds 2abcosC instead of subtracting it. Correct the rule and find c.3 marks
Cosine rule subtracts 2abcosC.
cos110° is negative, so subtracting this term increases c².
c²=117−108cos110°; c≈12.4 units.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗