Trigonometry

Overview

Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has since been generalized to refer to the study of periodicity.

Angles

An angle is formed by rotating a ray about its initial point. The ray in its initial position is called the initial side of the angle, and the position of the ray after it has been rotated is called the terminal side of the angle. The initial point of the ray is called the vertex of the angle.

When the vertex of an angle is at the origin in the Cartesian coordinate system, and the initial side lies along the positive $x$-axis, then the angle is in standard position.

Subtended Angles

An angle subtended by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. An angle subtended by an arc of a curve is the angle subtended by the corresponding chord of the arc.

Measurements

Angles are measures in degrees and radians.

An angle of $1\mathrm{°}$ is the angle in standard position on the unit circle that is subtended by an arc measuring $1/360$th of the circle's circumference. An angle of $1$ radian is the angle in standard position on the unit circle that is subtended by an arc measuring the length of the circle's radius (in the positive direction).

On a circle of radius $r$, the arc length $s$ intercepted by a central angle with radian measure $\theta$ is $s=r\theta$.

Notice then that a radian measure is a ratio of two lengths (the quotient of the length of an arc by a radius of a circle), which makes radian measure a dimensionless quantity.

Triangles

An angle measuring less than $90\mathrm{°}$ is called an acute angle. An angle measuring exactly $90\mathrm{°}$ is called a right angle. An angle measure greater than $90\mathrm{°}$ is called an obtuse angle.

Two angles that together sum up to $90\mathrm{°}$ are called complementary angles. Two angles that together sum up to $180\mathrm{°}$ are called supplementary angles.

Sinusoidal Functions

Let $A$, $B$, $C$, and $D$ be constants with $B\ne 0$. A sinusoidal function is one of form

The graph of a sinusoidal function is called a sinusoidal wave (or sinusoid). Both $y=\sin (t)$ and $y=\cos (t)$ are sinusoidal waves.

The amplitude of a sinusoid is one-half the distance between the minimum and maximum functional values. That is,

A horizontal shift of a sinusoid is called a phase shift.

Frequency

The frequency of a sinusoidal function is the number of periods (or cycles) per unit time. That is,