Law of Sines and Cosines Practice

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. It can be expressed as: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$ .

In the Law of Sines, side 'a' is the length of the side opposite angle 'A'.

In the Law of Sines, side 'b' is the length of the side opposite angle 'B'.

To find a missing side, set up the equation as follows: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$$ , where 'a' and 'b' are the known sides and 'A' and 'B' are the angles opposite those sides.

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is expressed as: $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$ .

Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS) of a triangle.

To find an angle using the Law of Cosines, rearrange the formula: $$C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$$ .

A triangle is possible if the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem).

The ambiguous case occurs when two sides and a non-included angle (SSA) are known, leading to potentially two different triangles, one triangle, or no triangle.

The area of a triangle can be found using the formula: $$Area = \frac{1}{2}ab \sin(C)$$ , where 'a' and 'b' are two sides and 'C' is the included angle.