Notes
Solving triangles
- Law of Sines: $\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Use it with AAS, ASA and SSA. SSA can be ambiguous (0, 1 or 2 triangles).
- Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$. Use it with SAS and SSS.
- Area: $\displaystyle \frac{1}{2}ab\sin C$, or Heron's formula $\displaystyle \sqrt{s(s-a)(s-b)(s-c)}$.
Vectors
- $\displaystyle |\langle a, b \rangle| = \sqrt{a^2 + b^2}$. A vector of magnitude $r$ at angle $\theta$ is $\langle r\cos\theta, r\sin\theta \rangle$.
- Dot product: $\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$. It is 0 for perpendicular vectors, and $\displaystyle \cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle c^2 = 25 + 25 - 2(25)\left(-\frac{1}{2}\right) = 75$.
$\displaystyle b = \frac{6\sin 45^\circ}{\sin 60^\circ} = \frac{6 \cdot \sqrt{2}/2}{\sqrt{3}/2} = \frac{6\sqrt{2}}{\sqrt{3}} = 2\sqrt{6}$.
$s = 21$, so area $\displaystyle = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84$.
$\displaystyle \sqrt{64 + 36} = 10$.
$12 - 10 = 2$.
$\displaystyle \frac{1}{2}(4)(6)\sin 150^\circ = 12 \cdot \frac{1}{2} = 6$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Triangle $ABC$ has sides $a = 7$, $b = 8$ and $c = 9$.
(a) Find $\cos C$ (the angle opposite side $c$).
Show the answer and scoring guide
$\displaystyle \cos C = \frac{49 + 64 - 81}{2 \cdot 7 \cdot 8} = \frac{32}{112} = \frac{2}{7}$.
Check off the points you earned:
(b) Find the area using Heron's formula.
Show the answer and scoring guide
$s = 12$, so area $\displaystyle = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = 12\sqrt{5}$.
Check off the points you earned:
(c) Confirm the area using $\frac{1}{2}ab\sin C$.
Show the answer and scoring guide
$\displaystyle \sin C = \sqrt{1 - \frac{4}{49}} = \frac{\sqrt{45}}{7} = \frac{3\sqrt{5}}{7}$. Area $\displaystyle = \frac{1}{2}(7)(8)\cdot\frac{3\sqrt{5}}{7} = 12\sqrt{5}$.
Check off the points you earned: