Finding Sides and Angles of Triangles

ASA, SAA, SSA
$$ \frac{a}{\sin{(a)}} = \frac{b}{\sin{(b)}} = \frac{c}{\sin{(c)}}$$
SAS, SSS
$$ a^2 = b^2 + c^2 -2bc\cos(A) $$
$$ b^2 = a^2 + c^2 -2ac \cos(B) $$
$$ c^2 = a^2 + b^2 -2ab \cos(C) $$
Double check
$$ \frac{a + b}{c} = \frac{\sin(A) + \sin(B)}{\sin(C)} $$

Ambiguous Case: SSA

Ambiguous Case SSA Table
Opposite > Adjacent Opposite = Adjacent Opposite < Adjacent
$$ \theta > 90^\circ $$ 1 Triangle 0 Triangles 0 Triangles
$$ \theta = 90^\circ $$ 1 Triangle 0 Triangles 0 Triangles
$$ \theta < 90^\circ $$ 1 Triangle 1 Triangle ***0, 1, or 2 noncongruent Triangles***
$$ \sin{\theta} = \frac{opp}{"adj"} $$
IF $$ opp = "adj"\sin{\theta} \therefore 1\triangle $$
IF $$ opp < "adj"\sin{\theta} \therefore 0\triangle $$
IF $$ opp > "adj"\sin{\theta} \therefore 2\triangle $$

Area

Generic Triangle Area Formulas
$$ T = \frac{1}{2}bh $$
$$ T = \frac{1}{2}l_1l_2 $$
$$ t = \sqrt{s(s-a)(s-b)(s-c)} $$
where $$ p = a + b + c $$
$$ s = \frac{1}{2}p $$
SAS Area Formula
$$ T = \frac{1}{2}s_1s_2\sin{\theta} $$
ASA Area Formula
$$ T \frac{s^2}{2} \times \frac{1}{\cot{\theta} + \cot{\phi}} $$

Vector / Scalar Comment

Vector
$$ \boldsymbol{R}, \vec{R}, \bar{R} $$
Scalar
Magnitude of Vector
$$ |\boldsymbol{R}|, |\vec{R}|, |\bar{R}| $$
Vector addition = (finds) Resultant