Parallelogram Law | Definitions and Laws of Vector Algebra

**Review of Plane Trigonometry:**
The parallelogram law is a powerful tool, but we don't want to rely on graphical measurements to determine the magnitude and
direction of the vector resultant. What we would like to do is to use trigonometry instead. In this section, we will review the
basics of trigonometry and the corresponding laws that are commonly used in Statics.

**Law of Sines:** For an arbitrary plane triangle of sides
*a, b, c*and interior angles α, β, γ as shown below, the law of sines is expressed as

(1) |

For example, if the values of |

**Law of Cosines:** For the same arbitrary triangle of
sides a, b, c and interior angles α, β, γ, the law of cosines is expressed as

(2) |

If we know the lengths of two sides and the magnitude of the angle in between, then we can solve for the length of the third side using the above relationship.

**Special Case of a Right Triangle:** If one of the three angles (for instance g)
is 90°, we will have a right triangle. In this case, Eq. (1) reduces to

; and Eq. (2) to |

We also have the following trigonometric relations for the right triangle

; | ; | ; |