Law of Sines / Law of Cosines

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: $$\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}$$ .

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: $$c^2 = a^2 + b^2 - 2ab \text{cos}(C)$$ .

To find a missing side using the Law of Sines, use the formula: $$\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)}$$ , where 'a' and 'b' are the sides and 'A' and 'B' are the angles opposite those sides.

To find a missing angle using the Law of Sines, rearrange the formula: $$\text{sin}(A) = \frac{a \times \text{sin}(B)}{b}$$ and use the inverse sine function.

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

To find a side using the Law of Cosines, use the formula: $$c^2 = a^2 + b^2 - 2ab \text{cos}(C)$$ and solve for 'c'.

To find an angle using the Law of Cosines, rearrange the formula: $$\text{cos}(C) = \frac{a^2 + b^2 - c^2}{2ab}$$ and use the inverse cosine function.

In a triangle, the larger the angle, the longer the side opposite that angle. This is a key principle in both the Law of Sines and the Law of Cosines.

The Law of Sines is particularly useful for solving non-right triangles, allowing for the calculation of unknown sides and angles when given sufficient information.

The Law of Cosines is essential for solving triangles when two sides and the included angle are known, or when all three sides are known.