The sine and cosine rules
| English | Chinese | Pinyin |
|---|---|---|
| sine rule | 正弦定理 | zhèng xián dìng lǐ |
| cosine rule | 余弦定理 | yú xián dìng lǐ |
| included angle | 夹角 | jiā jiǎo |
When the triangle isn't right-angled
- Pythagoras and SOH-CAH-TOA only work for right-angled triangles.
- For any other triangle, you need the sine rule 正弦定理 or the cosine rule 余弦定理.
The sine rule
- For any triangle with sides $a, b, c$ opposite angles $A, B, C$:
- Use it when you know: a side and its opposite angle, plus one other piece.
$a = 10$, $A = 30^{\circ}$, $B = 50^{\circ}$: $\;\dfrac{10}{\sin 30^{\circ}} = \dfrac{b}{\sin 50^{\circ}} \Rightarrow b = \dfrac{10 \sin 50^{\circ}}{0.5} = 15.3$.
$y=\sin x$ and $y=\cos x$ are smooth waves between $-1$ and $1$; cosine is sine shifted left by $90^\circ$
Sine & cosine rule
Two sides and the angle between them fix the triangle: the cosine rule finds the third side, the sine rule the other angles.
What sin and cos mean
Spin the angle on the unit circle: the horizontal leg is cos θ and the vertical leg is sin θ — the same ratios the sine and cosine rules use.
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.
The sine rule requires a side and its opposite angle to be known.
You need at least one side paired with its opposite angle to apply the sine rule.
In a triangle, a = 10, A = 30°, B = 50°. Using the sine rule, b = 10 sin 50° / sin 30°. Find b (1 dp).
b = 10 × sin 50° / 0.5 = 10 × 0.766 / 0.5 = 15.3.
The cosine rule
- Use it when you know: two sides and the included angle 夹角 (to find the third side), or all three sides (to find an angle).
Included angle only. The angle $A$ in the cosine rule must be the angle between sides $b$ and $c$. If it's not the included angle, the formula doesn't apply directly.
In any triangle, sides $a,b,c$ lie opposite angles $A,B,C$; the sine and cosine rules use these
A triangle has b = 7, c = 8 and the angle between them A = 40°. Using a² = 49 + 64 − 2(7)(8)cos40° ≈ 27.2, find a (1 dp).
a = √27.2 ≈ 5.2 cm.
In the cosine rule a² = b² + c² − 2bc cos A, the angle A must be the ______ angle between sides b and c.
A must be the included angle (the angle between sides b and c).
Area of any triangle
- This works for any triangle (not just right-angled ones), using two sides and the included angle.
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.
Worked example
- $b = 7$, $c = 8$, included angle $A = 40^{\circ}$.
- $a^2 = 7^2 + 8^2 - 2(7)(8)\cos 40^{\circ} = 49 + 64 - 85.8 = 27.2$, so $a = 5.2$ cm.
The cosine rule generalises Pythagoras: when $A = 90^{\circ}$, $\cos A = 0$ and you get $a^2 = b^2 + c^2$.
You've got it
- 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 $= \dfrac{1}{2}ab\sin C$