Learning Objectives
  • Use the sine rule and cosine rule in non-right-angled triangles.
  • Choose the correct rule from the information given.
  • Calculate the area of a triangle using two sides and the included angle.
  • Handle obtuse angles and the ambiguous sine-rule case.
  • Give angles to the accuracy expected by Cambridge.
Key Terms
Sine rule
A relationship between side lengths and sines of opposite angles.
Cosine rule
A relationship involving three sides and one angle of any triangle.
Included angle
The angle between two specified sides.
Ambiguous case
A sine-rule situation that may produce two possible triangles.
Opposite pair
A side and the angle directly opposite it.
Bearing
A clockwise angle from north, written with three figures.
What This Chapter Covers
  • Sine rule for sides and angles
  • Cosine rule for sides and angles
  • Triangle area using \frac12ab\sin C
  • Obtuse angles and ambiguous cases
  • Rule selection and calculator accuracy
Labelling A Triangle

In triangle ABC, side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. Correct opposite pairing is essential. A diagram should be labelled before choosing a formula.

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
a^2=b^2+c^2-2bc\cos A
\text{Area}=\frac12ab\sin C
Choosing The Rule
Information available Typical choice
One known opposite side–angle pair plus another side or angle Sine rule
Three sides and an angle required Cosine rule
Two sides and the included angle, third side required Cosine rule
Two sides and included angle, area required \frac12ab\sin C
Right-angled triangle Usually basic sine, cosine, tangent or Pythagoras instead
Sine Rule For A Side
Worked Example: Find A Side

Question: In triangle ABC, A=42^\circ, B=71^\circ and a=8.4 cm. Find b.

  1. Pair a with A and b with B.
  2. \frac{b}{\sin71^\circ}=\frac{8.4}{\sin42^\circ}.
  3. Rearrange: b=8.4\frac{\sin71^\circ}{\sin42^\circ}.
  4. Use degree mode and round only at the end.

Answer: b\approx11.9 cm.

Sine Rule For An Angle
Worked Example: Find An Angle

Question: In triangle PQR, p=9.2, q=12.5 and P=38^\circ. Find Q.

  1. \frac{\sin Q}{12.5}=\frac{\sin38^\circ}{9.2}.
  2. \sin Q=\frac{12.5\sin38^\circ}{9.2}\approx0.8365.
  3. First solution Q\approx56.8^\circ.
  4. Check the ambiguous alternative 180^\circ-56.8^\circ=123.2^\circ.
  5. Both are possible only if the remaining angle is positive and the context allows both triangles.

Answer: Possible Q=56.8^\circ or 123.2^\circ, subject to the diagram and conditions.

The Ambiguous Case

Because \sin\theta=\sin(180^\circ-\theta), inverse sine may give an acute angle while an obtuse angle is also possible. This issue occurs when two sides and a non-included angle are known. After obtaining the first angle, test its supplement and check the triangle angle sum.

Ambiguous-Case Check
  1. Use the sine rule to find the first angle \theta.
  2. Calculate the alternative 180^\circ-\theta.
  3. Add each candidate to the known angle.
  4. Discard any candidate that makes the angle sum at least 180^\circ.
  5. Use diagram or context restrictions to decide whether one or two triangles are valid.
Cosine Rule For A Side
Worked Example: Two Sides And Included Angle

Question: Two sides of a triangle are 7.5 cm and 11.2 cm, with included angle 64^\circ. Find the third side x.

  1. Use x^2=7.5^2+11.2^2-2(7.5)(11.2)\cos64^\circ.
  2. Calculate the right-hand side without premature rounding.
  3. Take the positive square root.

Answer: x\approx10.4 cm.

Cosine Rule For An Angle
Worked Example: Three Sides

Question: A triangle has sides 6 cm, 9 cm and 11 cm. Find the angle C opposite the 11 cm side.

  1. Use 11^2=6^2+9^2-2(6)(9)\cos C.
  2. Rearrange: \cos C=\frac{6^2+9^2-11^2}{2(6)(9)}.
  3. Apply inverse cosine.

Answer: C\approx92.1^\circ.

Area Of A Non-Right-Angled Triangle
Worked Example: Area

Question: Find the area of a triangle with sides 13 cm and 17 cm enclosing an angle of 48^\circ.

  1. Use \frac12ab\sin C.
  2. \text{Area}=\frac12(13)(17)\sin48^\circ.

Answer: \approx82.1\text{ cm}^2.

Obtuse Angles

The cosine of an obtuse angle is negative. The cosine rule handles this automatically. Keep the negative sign generated by the calculator and use degree mode. In a triangle, the longest side is opposite the largest angle, which provides a useful reasonableness check.

Examination Guidance
  • Label opposite side–angle pairs before substituting.
  • Use the cosine rule for SAS or SSS information.
  • Use the sine rule only when a complete opposite pair is known.
  • Check for a second sine-rule angle in the ambiguous case.
  • Give angle answers to one decimal place unless instructed otherwise and keep full calculator values during working.
Common Mistakes
  • Pairing a side with an adjacent rather than opposite angle.
  • Using the sine rule when no opposite pair is known.
  • Forgetting the square root after using the cosine rule to find a side.
  • Ignoring the obtuse alternative after inverse sine.
  • Using radians instead of degrees.
Knowledge Check And Practice

1. Which rule is most suitable when all three sides are known and one angle is required?

Answer: Cosine rule.

2. Which rule is suitable when two angles and one side are known?

Answer: Sine rule.

3. State the area formula using sides a and b and included angle C.

Answer: \frac12ab\sin C.

4. Find c if a=5, b=8, C=60^\circ.

Answer: c=7.

5. Why can the sine rule produce two possible angles?

Answer: Because \sin\theta=\sin(180^\circ-\theta).

6. What side is opposite the largest angle in a triangle?

Answer: The longest side.
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