Grasp Maths

Year 10

Sine rule and cosine rule where appropriate

Select and apply the sine rule and cosine rule to find unknown sides and angles in non-right-angled triangles, and justify which method is most efficient.

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Lesson overview

Geometry — trigonometry for non-right triangles

For any triangle with sides a, b, c and opposite angles A, B, C: **Sine rule:** asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. Use this to find a side if two angles and one side are known (AAS or ASA), or to find an angle if two sides and an angle opposite one of them are known (SSA, with care about the ambiguous case). **Cosine rule:** a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A. Use this to find a side if two sides and the included angle are known (SAS), or to find an angle if all three sides are known (SSS). The cosine rule is also used to find the third side when two sides and a non-included angle are known.

Using the sine rule to find a side (AAS)

In a triangle, angle A = 40°, angle B = 75° and side a = 8 cm. Find side b.

a = 8 cmb = ?A = 40°B = 75°

When you know two angles and a side, use the sine rule. First find the third angle if needed (angles sum to 180°).

Using the sine rule to find an angle (SSA)

In a triangle, side a = 10 m, side b = 8 m and angle A = 50°. Find angle B.

a = 10 mb = 8 mA = 50°B = ?

SSA can give two solutions. Check which is valid by considering whether all angles sum to 180°.

Using the cosine rule to find a side (SAS)

In a triangle, a = 5 cm, c = 7 cm and angle B = 60°. Find side b.

c = 7 cmb = ?a = 5 cmB = 60°

When you have two sides and the included angle, use the cosine rule to find the opposite side.

Using the cosine rule to find an angle (SSS)

In a triangle with sides a = 3, b = 4, c = 5, find angle C.

a = 3c = 5b = 4C = ?

When you know all three sides, use the cosine rule rearranged to find the angle.

Worked example — choosing the right method

In a triangle, sides are a = 7 cm and b = 5 cm, and angle A = 55°. Find angle B and then side c.

a = 7 cmb = 5 cmA = 55°B = ?
  1. Identify what you know: two sides (a = 7, b = 5) and a non-included angle (A = 55°). This is SSA.
  2. Use the sine rule to find angle B: asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}.
  3. Solve: sin⁡B=5sin⁡55°7≈0.586\sin B = \frac{5 \sin 55°}{7} ≈ 0.586, so B≈35.9°B ≈ 35.9°.
  4. Check: could there be another solution? 180°−35.9°=144.1°180° - 35.9° = 144.1° would give A + B > 180°, so only one valid triangle.
  5. Find the third angle: C=180°−55°−35.9°=89.1°C = 180° - 55° - 35.9° = 89.1°.
  6. Use the sine rule again to find side c: c=7sin⁡89.1°sin⁡55°≈8.5c = \frac{7 \sin 89.1°}{\sin 55°} ≈ 8.5 cm.
  7. Final answer: angle B ≈ 35.9°, angle C ≈ 89.1°, side c ≈ 8.5 cm.

Try it

Identify what you know about the triangle (sides and angles). Choose the sine rule for AAS/ASA/SSA or the cosine rule for SAS/SSS.

Question 1

In a triangle, angle A = 50°, angle B = 70° and side a = 6 cm. Use the sine rule to find side b.

a = 6 cmb = ?A = 50°B = 70°

💡 6sin⁡50°=bsin⁡70°\frac{6}{\sin 50°} = \frac{b}{\sin 70°}. Solve: b=6sin⁡70°sin⁡50°b = \frac{6 \sin 70°}{\sin 50°}.

Question 2

In a triangle, sides a = 5, c = 8 and angle B = 45°. Use the cosine rule to find side b.

c = 8b = ?a = 5B = 45°

💡 b2=52+82−2(5)(8)cos⁡45°b^2 = 5^2 + 8^2 - 2(5)(8) \cos 45°.

Question 3

In a triangle with sides a = 3, b = 4, c = 5, find angle C using the cosine rule.

a = 3c = 5b = 4C = ?

💡 52=32+42−2(3)(4)cos⁡C5^2 = 3^2 + 4^2 - 2(3)(4) \cos C. Simplify to find cos⁡C=0\cos C = 0.

Question 4

In a triangle, side a = 7, angle A = 40° and angle B = 60°. Find side b using the sine rule.

a = 7b = ?A = 40°B = 60°

💡 7sin⁡40°=bsin⁡60°\frac{7}{\sin 40°} = \frac{b}{\sin 60°}.

Question 5

When should you use the cosine rule instead of the sine rule?

💡 SAS (two sides and included angle) and SSS (all three sides) require the cosine rule.

Common mistakes

Watch for these when working through the lesson.

  • Using the sine rule for SAS or SSS. Remember: SAS and SSS require the cosine rule.
  • Forgetting that the SSA case can have two solutions (the ambiguous case). Always check whether a second solution is valid.
  • Mixing up the formula. Cosine rule: a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A. Make sure you use the side opposite the angle you're looking for.
  • Not rearranging the cosine rule correctly when finding an angle. From a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A, we get cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}.

Related topics

These ideas fit closely with this lesson.

  • Trigonometry in right-angled triangles
  • Bearings and navigation
  • Area of a triangle using $\frac{1}{2}ab \sin C$

Practice next

Independent practice will plug in here

This lesson builds the understanding first. Deeper adaptive practice can sit here later.