Precalculus / Unit 6: Applications of Trigonometry
6.1 The 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 for AAS, ASA and SSA (two angles and a side, or two sides and a non-included angle).
- The angles of a triangle add to $180^\circ$.
- Ambiguous case (SSA): there may be 0, 1 or 2 triangles. If $\sin B$ comes out greater than 1 there is no triangle.
- Example: $A = 30^\circ$, $B = 45^\circ$, $a = 10$ gives $\displaystyle b = \frac{10\sin 45^\circ}{\sin 30^\circ} = \frac{10 \cdot \sqrt{2}/2}{1/2} = 10\sqrt{2}$.
Practice
In a triangle, $A = 30^\circ$, $B = 45^\circ$ and $a = 10$. What is $b$?
$\displaystyle b = \frac{a\sin B}{\sin A} = \frac{10 \cdot \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2}$.
In a triangle, $A = 40^\circ$ and $B = 75^\circ$. What is $C$?
$180^\circ - 40^\circ - 75^\circ = 65^\circ$.
A triangle has $a = 8$, $b = 10$ and $A = 30^\circ$. How many such triangles exist?
$\displaystyle \sin B = \frac{10 \cdot \frac{1}{2}}{8} = \frac{5}{8} < 1$, and $b > a$, so both an acute and an obtuse $B$ work.
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