Integrated Math 3 / Unit 6: Trigonometric Identities and Laws
6.3 Law of Sines
Notes
In any triangle with sides $a, b, c$ opposite angles $A, B, C$: $\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
- Use it when you know two angles and a side (AAS or ASA), or two sides and an angle not between them (SSA).
- Angles in a triangle add to $180^\circ$, so find the third angle first.
- Ambiguous case (SSA): with angle $A$, side $a$ opposite and side $b$, let $h = b\sin A$. If $a < h$ there is no triangle. If $a = h$ there is one right triangle. If $h < a < b$ there are two triangles. If $a \geq b$ there is one.
- Example: $A = 30^\circ$, $B = 45^\circ$, $a = 8$ gives $\displaystyle b = \frac{8\sin 45^\circ}{\sin 30^\circ} = 8\sqrt{2} \approx 11.3$.
Practice
In a triangle, $A = 30^\circ$, $B = 45^\circ$ and $a = 8$. What is $b$ to the nearest tenth?
$\displaystyle b = \frac{8\sin 45^\circ}{\sin 30^\circ} = \frac{8(0.7071)}{0.5} \approx 11.3$.
In a triangle, $A = 50^\circ$, $B = 70^\circ$ and $c = 12$. What is $a$ to the nearest tenth?
$C = 180^\circ - 50^\circ - 70^\circ = 60^\circ$. Then $\displaystyle a = \frac{12\sin 50^\circ}{\sin 60^\circ} \approx 10.6$.
A triangle has $A = 30^\circ$, $a = 7$ and $b = 10$. How many triangles are possible?
$h = 10\sin 30^\circ = 5$. Since $5 < 7 < 10$, there are two triangles.
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