Introduction to Trigonometric Functions and the Unit Circle

Foundations of Trigonometric Functions

• Trigonometric functions relate a central angle to the ratio of two sides of a right triangle.

• Similar Triangles: The ratio between sides (e.g., $\frac{a}{b}$) for a particular angle is unchanging regardless of the triangle's size. This makes trigonometric ratios predictable and repeatable.

• Relationships are defined by an angle and two sides.

Right Triangle Trigonometry Definitions

• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$

The Unit Circle and Arc Length

• The unit circle is a circle with a radius of $1$, typically superimposed on an $xy-axis$.

• Arc Length ($t$): Measured counterclockwise (positive) or clockwise (negative) from the point $(1, 0)$.

• Central Angle ($\theta$): Each arc length identifies a unique point $(x, y)$ and a unique angle.

• Numerical Identity: According to the formula $s = r \times \theta$, when $r = 1$, the arc length $s$ (or $t$) is numerically equal to the central angle $\theta$ measured in radians.

Trigonometric Definitions on the Unit Circle

For a point $(x, y)$ on the unit circle:

• $\sin(t) = y$

• $\cos(t) = x$

• $\tan(t) = \frac{y}{x}$ ($x \neq 0$)

• Reciprocal Functions:

  • $\csc(t) = \frac{1}{y}$ (Reciprocal of sine)

  • $\sec(t) = \frac{1}{x}$ (Reciprocal of cosine)

  • $\cot(t) = \frac{x}{y}$ (Reciprocal of tangent)

Quadrant Angles (Quadrantal Angles)

• $0$ radians ($0^{\circ}$) at $(1, 0)$: $\sin(0) = 0$, $\cos(0) = 1$, $\tan(0) = 0$. Reciprocals: $\csc(0)$ and $\cot(0)$ are $\text{undefined}$.

• $\frac{\pi}{2}$ radians ($90^{\circ}$) at $(0, 1)$: $\sin(\frac{\pi}{2}) = 1$, $\cos(\frac{\pi}{2}) = 0$, $\tan(\frac{\pi}{2})$ is $\text{undefined}$, $\cot(\frac{\pi}{2}) = 0$.

• $\pi$ radians ($180^{\circ}$) at $(-1, 0)$: $\sin(\pi) = 0$, $\cos(\pi) = -1$, $\tan(\pi) = 0$.

• $\frac{3\pi}{2}$ radians ($270^{\circ}$) at $(0, -1)$: $\sin(\frac{3\frac{\pi}{2}}) = -1$, $\cos(\frac{3\pi}{2}) = 0$, $\tan(\frac{3\pi}{2})$ is $\text{undefined}$, $\cot(\frac{3\pi}{2}) = 0$.

Example: Point in Quadrant II

Given a point $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle:

• Sine and Cosine: $\sin(\theta) = \frac{\sqrt{3}}{2}$ (positive in Q2), $\cos(\theta) = -\frac{1}{2}$ (negative in Q2).

• Tangent: $\tan(\theta) = \frac{\sqrt{3}/2}{-1/2} = -\sqrt{3}$.

• Reciprocals:

  • $\csc(\theta) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

  • $\sec(\theta) = -2$

  • $\cot(\theta) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$