Question

question 15 of 20
which of the following problem types can always be solved using the law of cosines or sines? check all that apply.
a. ssa
b. aaa
c. asa
d. sas
e. saa
f. sss

Explanation:

• SSA (Side - Side - Angle): The Law of Sines can be used. However, there may be the ambiguous case (0, 1, or 2 solutions). But it is still solvable using the Law of Sines (and sometimes the Law of Cosines for verification).
• ASA (Angle - Side - Angle): We know two angles and the included side. First, we find the third angle using the fact that the sum of angles in a triangle is \(180^{\circ}\). Then we use the Law of Sines (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)) to find the other two sides.
• SAS (Side - Angle - Side): We use the Law of Cosines (\(c^{2}=a^{2}+b^{2}-2ab\cos C\)) to find the third side and then the Law of Sines to find the other angles.
• SAA (Side - Angle - Angle): This is equivalent to ASA (since if we know two angles, we know the third angle). We use the Law of Sines.
• SSS (Side - Side - Side): We use the Law of Cosines (\(\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\), \(\cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac}\), \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)) to find the angles.
• AAA (Angle - Angle - Angle): We cannot find the side lengths uniquely. There are infinitely many similar triangles with the same set of angles.

Answer:

A. SSA, C. ASA, D. SAS, E. SAA, F. SSS