Rukasha Maths

GCSE maths Beta

Non-right-angled trigonometry · Revision guide

Higher · Geometry · 2 pages in the PDF

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Learning objectives

  • Select the sine rule, cosine rule or triangle area formula.
  • Match opposite sides and angles, and round final values only.

Key ideas

Use opposite side-angle pairs in the sine rule.

asin⁡A=bsin⁡B=csin⁡C\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}

The cosine rule uses the included angle between the two known sides.

a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A

The area formula uses the included angle.

area=12absin⁡C\text{area}=\frac12ab\sin C

Use degree mode for degree angles. A sine-rule angle may have two possible values; check the triangle context.

Worked examples

Example 1

Sides of 7 cm and 9 cm enclose 60 degrees. Find the third side and area to three significant figures.

c2=72+92−2(7)(9)cos⁡60∘=67c^2=7^2+9^2-2(7)(9)\cos60^\circ=67
c=67 cmc=\sqrt{67}\ \text{cm}
area=12(7)(9)sin⁡60∘=6334 cm2\text{area}=\frac12(7)(9)\sin60^\circ=\frac{63\sqrt3}{4}\ \text{cm}^2
c≈8.19 cm,area≈27.3 cm2c\approx8.19\ \text{cm},\qquad\text{area}\approx27.3\ \text{cm}^2

Example 2

An 8 cm side is opposite 30 degrees. Find the side opposite 60 degrees exactly.

8sin⁡30∘=bsin⁡60∘\frac8{\sin30^\circ}=\frac b{\sin60^\circ}
b=8sin⁡60∘sin⁡30∘=83 cmb=\frac{8\sin60^\circ}{\sin30^\circ}=8\sqrt3\ \text{cm}

Watch out

Pythagoras requires a right angle. Match each sine-rule side to its opposite angle; use the included angle for the cosine rule.