Trigonometry / Unit 8: Oblique Triangles and Vectors
8.2 The Ambiguous Case (SSA)
Notes
When you know two sides and an angle not between them, there can be 0, 1 or 2 triangles.
- Suppose $A$ is acute, with sides $a$ (opposite $A$) and $b$. Compute the height $h = b\sin A$.
- If $a < h$: no triangle. If $a = h$: one right triangle.
- If $h < a < b$: two triangles (one angle $B$ acute, the other obtuse).
- If $a \ge b$: one triangle.
- If $A$ is obtuse (or right), there is one triangle when $a > b$, otherwise none.
Practice
In a triangle $A = 30^\circ$, $a = 4$ and $b = 10$. How many triangles are possible?
$h = 10\sin 30^\circ = 5$, and $a = 4 < 5$, so side $a$ is too short to reach.
In a triangle $A = 30^\circ$, $a = 6$ and $b = 10$. How many triangles are possible?
$h = 10\sin 30^\circ = 5$ and $5 < 6 < 10$, so there are two triangles.
In a triangle $A = 40^\circ$, $a = 12$ and $b = 8$. How many triangles are possible?
Since $a \ge b$, side $a$ swings to only one position.
Ready for more? Take the Unit 8 test.