Geometry Lab reference
The law of sines
A shared ratio between sides and their opposite angles.
a / sin A = b / sin B = c / sin C
The idea
Use the sine rule when you know an angle and its opposite side, together with another side or angle. The ratio of a side to the sine of its opposite angle is the same throughout a triangle.
Work through an example
For A = 30°, B = 45°, and a = 5, b = 5 × sin 45° / sin 30° = 5√2 ≈ 7.071. The remaining angle is C = 180° − 30° − 45° = 105°.
What to watch for
SSA can have two answers because sin B = sin(180° − B). Both candidates must be checked against the 180° total. For a = 7, b = 10, A = 30°, two different triangles satisfy the inputs.
Make the formula move.
Explore this idea with real inputs.