I'll go through the objectives one by one and say something about them.

1. The Law of Sines

The Law of Sines is a very easy way of relating every side in a triangle, along with the angles.
Although sinA/a = sinB/b is equal to their reciprocal, it's better to do
a/sinA = b/sinB = c/sinc, because it also equals 2R, where R is the radius of the circumscribed circle

The proof of the Law of Sines is like so: given a triangle ABC, (counter-clockwise), you can draw an altitude, and label it X. on one side, b sin A = x, and on the other side a sin B = x, so a sin B = b sin A, so a/sinA = b/sin B (it's on khanacademy)

2. The Law of Cosines
Unlike the Law of Sines, which could yield the incorrect result (sin(x) = sin(180-x)), the Law of Cosines always works. It's kind of similar to the Pythagorean Theorem:
a^2 = b^2 + c^2 - 2bc*cos(A). So, when cos(A) = 0, at 90 degrees, it equals a^2 = b^2 + c^2, a being the hypotenuse.

Proof: Given the same triangle from above, you can draw an altitude going into side c. Now, assume you know angle A (bottom left). The adjacent side (part of C), is bcos(theta). The altitude is bsin(theta). The other part is C - bcos(theta). So the hypotenuse of the right side (a) squared equals the other two sides squared.
a^2 = (bsin(theta))^2 + (c- bcos(theta))^2
a^2 = b^2sin^2(theta) + c^2 - 2bccos(theta) + b^2cos^2theta
a^2 = b^2((sin^2(theta) + cos^2(theta)) + c^2 - 2bc*cos(theta)
a^2 = b^2 + c^2 - 2bc*cos(theta)

3. Using the Law of Sines and Cosines to solve triangles, including multi-step problems and word problems.
Just do a few problem

4. Problems that require some basic geometry knowledge (similar triangles, angle bisector theorem, properties of parallelograms, area of a sector of a circle, etc.)

Not much to it, just do it. (rhymes!)
Examples: #2, 4, 7, 8, 9 from the packet called "Using the Law of Sines and Cosines"

5. Proofs that use the Law of Cosines or Law of Sines

You should be able to prove, for example:

- The Triangle Inequality (use the Law of Cosines)

- The Pythagorean Inequalities (use the Law of Cosines)

(will do later)

6. Area Formulas:

- Area of a triangle = 1/2 ab sin(C)

- Heron's Formula - know the proof/derivation

- Using repeated applications of triangle solving to find the area of polygons by splitting them up into triangles: see Foerster, p. 261 #25 and 26

7.

The Ambiguous Case of the Law of Sines

see problem #2 from the quiz

8. Solving SSS and SAS triangles by being strategic about which angles to solve for first see problem #1 from the quiz and the discussion we had in class on this topic

9. The formulas for the radii of the circumscribed and inscribed circles of a triangle in terms of the sides and angles of a triangle -- Larson p. 507 #44 (a) and (b)

WILL FINISH COMMENTING ON TOPICS LATER