The sine and cosine rules
The sine and cosine rules (Extended)
- For any triangle with sides $a,b,c$ opposite angles $A,B,C$:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \qquad\text{(sine rule)}$$
$$a^2 = b^2 + c^2 - 2bc\cos A \qquad\text{(cosine rule)}$$
- Sine rule: a side with its opposite angle. Cosine rule: two sides and the angle between, or all three sides.
Practice
You know all three sides and want an angle. Which rule do you use?
With three sides (or two sides and the included angle), use the cosine rule.
Worked example + area
- $b=7$, $c=8$, angle between $A=40^{\circ}$: $a^2 = 49 + 64 - 2(7)(8)\cos 40^{\circ} = 27.2$, so $a = 5.2\ \text{cm}$.
- Area of any triangle: $\tfrac12 ab\sin C$.
Practice
A triangle has b = 7, c = 8 and the angle between them A = 40°. Using a² = 7² + 8² − 2(7)(8)cos40° = 27.2, find a (1 dp).
a = √27.2 = 5.2 cm.
Practice
The area of any triangle (two sides a, b with angle C between them) is:
Area = ½ab sin C uses two sides and the included angle.
You've got it
Key idea
- sine rule $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$ — side + its opposite angle
- cosine rule $a^2=b^2+c^2-2bc\cos A$ — two sides + included angle (or three sides)
- area of any triangle $= \tfrac12 ab\sin C$