Trigonometry / Unit 8: Oblique Triangles and Vectors
8.1 The Law of Sines
Notes
In any triangle with angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$:
- $\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$.
- Use it when you know an angle and its opposite side, plus one more piece: AAS or ASA (or SSA, see the next topic).
- Find the third angle first using $A + B + C = 180^\circ$.
- Example: $A = 30^\circ$, $B = 90^\circ$, $a = 5$ gives $\displaystyle b = \frac{5\sin 90^\circ}{\sin 30^\circ} = 10$.
Practice
In triangle $ABC$, $A = 40^\circ$, $B = 60^\circ$ and $a = 10$. What is $b$ to the nearest tenth?
$\displaystyle b = \frac{a\sin B}{\sin A} = \frac{10\sin 60^\circ}{\sin 40^\circ} \approx 13.5$.
Which set of given information can be started directly with the Law of Sines?
Two angles give the third, so you will know a side and its opposite angle. Three sides or two sides with the included angle call for the Law of Cosines. Three angles only give the shape, not the size.
In triangle $ABC$, $a = 12$, $A = 30^\circ$ and $B = 45^\circ$. What is the exact value of $b$?
$\displaystyle b = \frac{12\sin 45^\circ}{\sin 30^\circ} = \frac{12 \cdot \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 12\sqrt{2}$.
Ready for more? Take the Unit 8 test.