Precalculus - The Unit Circles

Introduction

A Unit Circle is a type of circle whose center is at $(0, 0)$ and whose radius length is equal to $1$ unit.
This circle is modeled by the equation $x^{2} + y^{2} = 1$ . This equation is derived using the distance formula.

The circumference of the unit circle is equal to $2 π$ units or approximately equal to $6.28$ units.
To inspect whether a point $(x, y)$ is on the unit circle, we check the coordinates if they satisfy the equation $x^{2} + y^{2} = 1$ .
The standard arc on the circle starts at the point $(1, 0)$ and travels counterclockwise if it is positive and clockwise if it is negative.
Since the circumference of the unit circle is $2 π$ , then the arc lengths are rational multiples of $2 π$ .
Consider the unit circle being divided by the coordinate axes into four congruent arcs.

As point X moves from point A counterclockwise and terminates at point B, the length of arc $A B$ is $\frac{1}{4}$ of $2 π$ or $\frac{π}{2}$ .
The length of arc $A B C$ is one-half of the circumference, which is $π$ .
The length of arc $A C D$ is $\frac{3}{4}$ of the circumference, which is $\frac{3 π}{2}$ .
If we consider θ as the length of the arc, then we have the following:
- $X (θ) = X (\frac{π}{2}) = (0, 1)$
- $X (θ) = X (π) = (- 1, 0)$
- $X (θ) = X (\frac{3 π}{2}) = (0, - 1)$
- $X (θ) = X (2 π) = (1, 0)$

As shown in the figure, each point on the unit circle satisfies the equation $x^{2} + y^{2} = 1$ .
Each point on the circle has coordinates corresponding to a special angle or multiple of special angles, such as $30 °, 60 °,$ and $90 °$ .
Study the tables below.

The domain of the unit circle $x^{2} + y^{2} = 1$ is $\{x | x \in ℝ, - 1 \leq x \leq 1\}$ . The range is $\{y | y \in ℝ, - 1 \leq y \leq 1\}$ .
Every real number (arc length) is associated uniquely with the central angle $θ$ that subtends the standard arc on the unit circle.
The relationship $P (θ + 2 k π) = P (θ)$ , for any integer $k$ is called the Periodic Theorem. Typical points on the unit circle range from $0 °$ to $360 °$ . Once the angle of the circular function exceeds $360 °$ , then this multiple of $360 °$ has to be subtracted from the given angle. The difference is then equivalent to one common value between $0 °$ to $360 °$ .

Let us consider a real number $θ$ and let $P (x, y)$ be a point on the unit circle corresponding to the angle $θ$ .
For each arc length on the unit circle, with the starting point $(1, 0)$ and terminal point $(x, y)$ , the values of $x$ and $y$ are real numbers, then we have $\cos (θ) = x$ and $\sin (θ) = y$ .
Study the figure below.

Point $P (x, y)$ is equal to $P (\cos (θ), \sin (θ))$ . This indicates that $\sin (θ) = y$ and $\cos (θ) = x$ . This is based on $\sin (θ) = \frac{y}{r} = \frac{y}{1} = y$ and $\cos (θ) = \frac{x}{r} = \frac{x}{1} = x$
Using the y and x coordinates, we can define the other trigonometric functions of $θ$ .
$\tan (θ) = \frac{y}{x}, x \neq 0$ , $c o t (θ) = \frac{x}{y}, y \neq 0,$ $s e c (θ) = \frac{1}{x}, x \neq 0,$ and $c s c (θ) = \frac{1}{y}, y \neq 0$ .
Trigonometric Functions are also called Circular Functions or Periodic Functions because they behave in a cyclic or repetitive manner.
To illustrate, we have $\sin (θ + 2 π) = \sin (θ)$ and $\cos (θ + 2 π) = \cos (θ)$ as these functions have a period of $2 π$ .

Trigonometric Functions of Any Angle

If we let $θ$ be any angle in standard position, it indicates that its vertex is at the origin and the initial side is the positive x-axis.
The six trigonometric functions are described in the tables below, and their respective signs in the four quadrants.

Reference Angle

For any angle $θ$ , its reference angle is the acute angle that the terminal side of $θ$ makes with the positive x-axis or negative x-axis.
If $0 ° < θ < 90 °$ , then the reference angle $β$ is equal to $θ$ .
In Quadrant II, the reference angle is $β = 180 ° - θ$ .
In Quadrant III, the reference angle is $β = θ - 180 °$ .
In Quadrant IV, the reference angle is $β = 360 ° - θ$ .

In general, the angle $θ$ and $(θ + 360 ° n)$ where $n \in ℤ$ have the same terminal side. These angles are called co-terminal angles.

Example 1. Find the exact value of $\cos (\frac{13 π}{6})$ using the standard unit circle.

Solution:

$\frac{13 π}{6} = 390 °$

$\frac{13 π}{6}$ terminates in Quadrant I, which forms $30 °$ in the positive x-axis.

Consider the angle $30 °$ and choose the point $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ , on the unit circle.

This shows that $\cos (\frac{13 π}{6}) = \cos (\frac{π}{6}) = \frac{\sqrt{3}}{2}$

Note that the sign is positive since $\cos (θ) > 0$ in Quadrant I.

Example 2. Given that $\tan (θ) = \frac{5}{12}$ and $\sin (θ) < 0$ , find the other trigonometric functions of $θ$ .

Solution:

If the tangent function is positive and sine function is negative, then $θ$ is Quadrant III.

$\tan (θ) = \frac{y}{x} = \frac{5}{12}$

Solve for the radius:

$r^{2} = \sqrt{x^{2} + y^{2}} \to r^{2} = \sqrt{{(12)}^{2} + {(5)}^{2}}$

$r = 13$

$\sin (θ) = - \frac{5}{13}, \cos (θ) = - \frac{12}{13}, c o t (θ) = \frac{12}{5}, s e c (θ) = - \frac{13}{12}, c s c (θ) = - \frac{13}{5}$

Trigonometric functions of different special angles are summarized in the tables below.

The cosine function and secant functions are even. To illustrate, we have $\cos (- θ) = \cos (θ)$ and $s e c (- θ) = s e c (θ)$ .
The sine, cosecant, tangent, and cotangent functions are odd.

Always be mindful of the sign of each trigonometric function in the four quadrants.
Reference angles can be used to evaluate trigonometric functions.
Suppose that $\tan (θ) = - \frac{1}{3}$ and we want to find the other trigonometric values of $θ$ , we solve for the radius. The radius is always positive.
The angles associated with trigonometry (in the unit circle approach) are directed angles. Angles obtained from the counterclockwise direction are positive while angles obtained from the clockwise direction are negative.