← LearnYear 10 Maths · Algebra & TrigonometryFree placement check
Year 10Extension

Sine Rule and Cosine Rule

Non-right-angled trigonometry is not part of the Year 10 core curriculum. Offered as preparation for senior mathematics.

Things you should know.

Method

Sine rule
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
Use when you have a matching side and angle pair.
Cosine rule
a2=b2+c2−2bccos⁡Aa^{2} = b^{2} + c^{2} - 2bc\cos A
Use for three sides, or two sides and the angle between them.
Area of a triangle
Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C
Two sides and the angle between them.

Worked example

e.g. find a if b=8, c=5, A=60∘\text{find } a \text{ if } b = 8,\ c = 5,\ A = 60^\circ
Triangle with sides 8 and 5 and a 60 degree angle between themA = 60°a = ?b = 8c = 5Two sides and the angle between them — cosine rule
  1. Decide which rule. Two sides and the angle BETWEEN them means cosine rulecosine rule\text{cosine rule}
  2. Write the formula with the unknown side as aaa2=b2+c2−2bccos⁡Aa^{2} = b^{2} + c^{2} - 2bc\cos A
  3. Substitutea2=82+52−2(8)(5)cos⁡60∘a^{2} = 8^{2} + 5^{2} - 2(8)(5)\cos 60^\circ
  4. Work out each parta2=64+25−80(0.5)=49a^{2} = 64 + 25 - 80(0.5) = 49
  5. Take the square roota=7a = 7
    The completed triangle, side a is 7A = 60°a = 7b = 8c = 5

Practice

Solve the following (round to 1 decimal place where needed):

1
b=8, c=5, A=60∘. Find ab = 8,\ c = 5,\ A = 60^\circ. \text{ Find } a
Answera=7a = 7
2
b=7, c=3, A=60∘. Find ab = 7,\ c = 3,\ A = 60^\circ. \text{ Find } a
Answera=37≈6.1a = \sqrt{37} \approx 6.1
3
a=5, b=7, c=8. Find Aa = 5,\ b = 7,\ c = 8. \text{ Find } A
AnswerA≈38.2∘A \approx 38.2^\circ
4
a=10, A=30∘, B=45∘. Find ba = 10,\ A = 30^\circ,\ B = 45^\circ. \text{ Find } b
Answerb≈14.1b \approx 14.1
5
a=12, A=40∘, B=70∘. Find ba = 12,\ A = 40^\circ,\ B = 70^\circ. \text{ Find } b
Answerb≈17.5b \approx 17.5
6
a=6, b=9, C=50∘. Find the areaa = 6,\ b = 9,\ C = 50^\circ. \text{ Find the area}
AnswerArea≈20.7\text{Area} \approx 20.7
7
a=9, b=12, c=15. Is it right angled?a = 9,\ b = 12,\ c = 15. \text{ Is it right angled?}
AnswerYes—92+122=225=152, so C=90∘\text{Yes} — 9^{2}+12^{2}=225=15^{2}, \text{ so } C = 90^\circ
8
b=4, c=6, A=120∘. Find ab = 4,\ c = 6,\ A = 120^\circ. \text{ Find } a
Answera=76≈8.7a = \sqrt{76} \approx 8.7
Next step
Practise Sine Rule and Cosine Rule with instant marking
A free 10-minute placement check finds which Year 10 topics to work on first, then Summit builds a weekly plan around them.
See where my child is →