Geometry

Point

A point is a location. It is represented as a dot, but has no dimensions. Below is a set of randomly plotted points.

Line

A line is an infinite set of points. Given any two points on a line, there is always a point that lies between them on that line. Below, the points ${A,}$ ${B,}$ ${C,}$ and ${D}$ are collinear. Points that do not lie on the same line are said to be collinear. We will denote lines with an overline double-ended arrow. For example, the line below can be denoted with the notation,

Line Segments

A line segment is a subset of a line. It comprises two distinct points on the line, called the line segment's endpoints, and every point in between. We denote a line segments with its two labeled endpoints, overlined. For example, the line above contains the segment ${\overline{BC}.}$

In the figure below, the point ${B}$ lies between ${A}$ and ${C}$ on ${\overline{AC}.}$ If ${\lvert \overline{AB} \rvert = \lvert \overline{BC} \rvert,}$ then ${B}$ is the midpoint of ${\overline{AC}.}$

Angles

An angle is a figure formed by two rays with a common endpoint called the vertex.

The symbol ${\theta}$ is often used to denote an angle. We can also label points on the angle's arms and vertex, and use those labels to name the angle. For example, we may refer to the angle above with the notation ${\angle ABC}$ (the middle letter is always reserved for the vertex).

Trigonometry

A triangle is a polygon with three edges (sides) and three vertices (corners). Thus, it is a set of three noncollinear points in a plane, with each pair of points connected by a line segment. These line segments form the triangle's sides. If the three noncollinear points are ${A,}$ ${B,}$ and ${C,}$ we denote their corresponding triangle with the notation ${\triangle ABC.}$

Two triangles are congruent if one coincides (fits perfectly over) with the other. For example, above, triangles ${\triangle ABC}$ and ${\triangle DEF}$ are congruent. We denote this with the notation ${\triangle ABC \cong \triangle DEF.}$

Trigonometric Ratios

The fundamental trigonometric ratios are sine, cosine, and tangent. These are ratios of the sides of a triangle, defined as follows (using the diagram below as reference):

definition. Suppose ${A,}$ ${B,}$ and ${C}$ form the sides of a triangle.

We define the trigonometric ratios of such a triangle as follows:

$$\cos \theta = \dfrac{\text{length of side adjacent to } \theta}{\text{length of hypotenuse}} = \dfrac{B}{C} \\[2em] \sin \theta = \dfrac{\text{length of side opposite to } \theta}{\text{length of hypotenuse}} = \dfrac{A}{C} \\[2em] \tan \theta = \dfrac{\text{length of side opposite to } \theta}{\text{length of side adjacent to } \theta} = \dfrac{A}{B} \\[2em] \sec \theta = \dfrac{\text{length of hypotenuse}}{\text{length of side adjacent to } \theta} = \dfrac{C}{B} \\[2em] \csc \theta = \dfrac{\text{length of hypotenuse}}{\text{length of side opposite to } \theta} = \dfrac{C}{A} \\[2em] \cot \theta = \dfrac{\text{length of side adjacent to } \theta}{\text{length of side opposite to } \theta} = \dfrac{B}{A}$$

From the definitions, we see there are three pairs of reciprocal functions:

Trigonometric Identities

The following are some common trigonometric identities. These identities are used extensively in simplifying trigonometric expressions.

Radian Measure

definition. Let ${\theta}$ be an angle with its vertex at the center of a circle of radius ${r.}$ Let ${s}$ denote the length of the arc intercepted by the angle. The radian measure of ${\theta}$ is the ratio of the arc length ${s}$ to the radius ${r.}$

$$m(\theta) = \dfrac{s}{r}.$$

One way to think about the radian measure: It is a fraction of the circumference of a circle. Recall that a circle's circumference is given by:

If the circle has a radius of ${1}$ (i.e., the radius of the unit circle), then the circumference is ${2\pi.}$ Thus, half the circle's circumference is ${\pi,}$ a quarter of the circle's circumference is ${\pi / 2,}$ ad infinitum. These fractional parts of the circle's circumference are called arcs. If we draw lines from the endpoints of these arcs to the circle's center, we get an angle between those lines. Thus, one way to measure the angle is by using the length of its corresponding arc. Hence:

As one might garner, this is a much more mathematically elegant way of measuring angles, as opposed to the arbitrary use of multiples of ${360 \degree.}$ Nevertheless, degrees are still useful in many contexts. As such, it's helpful to know how to convert between degrees and radians:

degrees-to-radians formula. Given a degree measure ${d,}$ the radian measure ${r}$ is given by:

$$r = d \left( \dfrac{\pi}{180}\right).$$

radians-to-degrees formula. Given a radian measure ${r,}$ the degree measure ${d}$ is given by:

$$d = r \left( \dfrac{180}{\pi}\right).$$

arc length formula. In a circle of radius ${r,}$ the arc length ${s}$ determined by a central angle of radian measure ${\theta}$ is given by

$$s = r \theta,$$

where ${s}$ and ${r}$ have the same linear units.

sector area formula. In a circle of radius ${r,}$ the area ${A}$ of a sector with central angle of radian measure ${\theta}$ is given by

$$A = \dfrac{1}{2} r^2 \theta.$$