Solving a triangle with the sine rule

When we know two angles and one side of a triangle, the sine rule lets us find the other sides.

Example — Solving a triangle, angles-side

A triangle has $\widehat{A} = 30°$, $\widehat{B} = 75°$, $a = 10$. Find $b$ and $c$.

First $\widehat{C} = 180° - 30° - 75° = 75°$. Hence the triangle is isosceles with $b = c$ (equal angles $\Rightarrow$ equal sides).

By the sine rule: $\frac{10}{\sin 30°} = \frac{b}{\sin 75°} \implies b = \frac{10\sin 75°}{\sin 30°} = 20\sin 75° \approx 19{,}32.$ And $c = b \approx 19{,}32$.

Links

Topics: Triangle trigonometry
Concepts: Triangle solving · Sine rule
Functions: Sine
Methods: Triangle solving
Skills: Calculating · Using formulae