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Applications of Trigonometry and Vectors

This page is meant to serve as a quick overview of the applications of trigonometry and vectors.

Law of Sines

The law of sines is used to find the angles of any triangle.
a / sin A = b / sin B = c / sin C

Area of a Triangle

A = (1/2) bc sinA
A = (1/2) ab sinC
A = (1/2) ac sinB

Four Cases for Oblique Triangles

1. Two angles and a side (AAS or ASA).
2. Two sides and an angle opposite one of them (SSA).
3. Three sides (SSS).
4. Two sides and the angle between them (SAS).

The Ambiguous Case of the Law of Sines

If A is acute:

If a < h, then there are no possible triangles.
If a = h, then there is only one possible triangle and that triangle is a right triangle.
If b > a > h, then there are two possible triangles.
Solve for the variable. Check the diagram and ensure the answer makes sense.

If A is obtuse:

If ab, then there are no possible triangles.
If a > b, then there is only one possible triangle.

Law of Cosines

The law of cosines is used to find the missing side or angle of a triangle.
= + - 2bc cos A
= + - 2ac cos B
= + - 2ab cos C

Heron's Area Formula

Given the lengths of the sides a, b, and c and the semiperimeter of a triangle, Heron's formula gives the area of the triangle.
Semipermiter: s = (1/2)(a + b + c)
Area of the triangle: A = √ (s (s - a) (s - b) (s - c))

Magnitude and Direction Angle of a Vector

The magnitude (lenth) of vector u = <a,b> is given by |u| = √( + ).
The direction angle Θ = arctan(b/a) where a ≠ 0.