Unit Circle Properties, Reference Numbers, and Terminal Points

Fundamentals of the Unit Circle

• A circle of radius $1$ ($r = 1$) centered at the origin $(0, 0)$ is defined as the unit circle.

• The unit circle represents the set of all points located at a distance of $1$ unit from the origin.

• Properties of the unit circle are foundational when studying and discussing trigonometric functions.

• The standard algebraic equation of the unit circle is:   $x^2 + y^2 = 1$

• Key axis-intercept points directly visible on the unit circle include:

  • Positive $x$-axis intercept: $(1, 0)$

  • Positive $y$-axis intercept: $(0, 1)$

  • Negative $x$-axis intercept: $(-1, 0)$

  • Negative $y$-axis intercept: $(0, -1)$

• To find coordinates of points on the unit circle that do not lie on the $x$-axis or $y$-axis, the equation $x^2 + y^2 = 1$ must be utilized directly along with given quadrant constraints.

Reference Numbers and Quadrant Rules

• Definition of Reference Number:

  • Let $t$ be a real number.

  • The reference number associated with $t$ is denoted as $\bar{t}$.

  • The reference number $\bar{t}$ represents the shortest distance along the perimeter of the unit circle between the terminal point determined by $t$ and the $x$-axis.

• Relationship Between Reference Numbers and Terminal Points:

  • Finding a terminal point in any quadrant only requires knowing the corresponding terminal point in the first quadrant.

  • Determining the quadrant in which the terminal point determined by $t$ lies is necessary to calculate its reference number $\bar{t}$.

• Quadrant-Specific Rules for Finding Reference Numbers $\bar{t}$:

  • First Quadrant (Quadrant I) and Fourth Quadrant (Quadrant IV):

  • In Quadrants I and IV, the $x$-coordinate of the terminal point is positive (x > 0).

  • The reference number $\bar{t}$ is found by measuring the shortest distance along the circle to the positive $x$-axis.

  • Second Quadrant (Quadrant II) and Third Quadrant (Quadrant III):

  • In Quadrants II and III, the $x$-coordinate of the terminal point is negative (x < 0).

  • The reference number $\bar{t}$ is found by measuring the shortest distance along the circle to the negative $x$-axis.

Step-by-Step Examples: Finding Coordinates and Reference Numbers

• Finding an Unknown Coordinate on the Unit Circle:

  • Problem Statement: A point lies on the unit circle in the third quadrant with an $x$-coordinate equal to $-\frac{\sqrt{3}}{2}$. Find its $y$-coordinate.

  • Step 1: Substitute the known $x$-coordinate $x = -\frac{\sqrt{3}}{2}$ into the unit circle equation $x^2 + y^2 = 1$:     $\left(-\frac{\sqrt{3}}{2}\right)^2 + y^2 = 1$

  • Step 2: Simplify the squared term:     $\frac{3}{4} + y^2 = 1$

  • Step 3: Subtract $\frac{3}{4}$ from both sides:     $y^2 = 1 - \frac{3}{4}$     $y^2 = \frac{1}{4}$

  • Step 4: Take the square root of both sides:     $y = \pm \frac{1}{2}$

  • Step 5: Apply quadrant rules to select the sign:

  • Since the point is located in the third quadrant, its $y$-coordinate must be negative.

  • Therefore, $y = -\frac{1}{2}$.

  • Complete Coordinates: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$

• Finding Reference Numbers for Specific Values of $t$:

  • Case 1: $t = \frac{5\pi}{6}$

  • The terminal point for $t = \frac{5\pi}{6}$ lies in the second quadrant.

  • The shortest distance along the circle to the $x$-axis is to the negative $x$-axis.

  • Calculation:       $\bar{t} = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$

  • Case 2: $t = \frac{7\pi}{4}$

  • The terminal point for $t = \frac{7\pi}{4}$ lies in the fourth quadrant.

  • The shortest distance along the circle to the $x$-axis is to the positive $x$-axis.

  • Calculation:       $\bar{t} = 2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$

• Determining Terminal Points Using Reference Numbers:

  • Finding Terminal Point for $t = \frac{5\pi}{6}$:

  • Reference number is $\bar{t} = \frac{\pi}{6}$.

  • In Quadrant I, $\bar{t} = \frac{\pi}{6}$ determines the terminal point $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

  • Since $t = \frac{5\pi}{6}$ lies in Quadrant II, its $x$-coordinate is negative (x < 0) and its $y$-coordinate is positive (y > 0).

  • Final desired terminal point: $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

  • Finding Terminal Point for $t = \frac{7\pi}{4}$:

  • Reference number is $\bar{t} = \frac{\pi}{4}$.

  • In Quadrant I, $\bar{t} = \frac{\pi}{4}$ determines the terminal point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

  • Since $t = \frac{7\pi}{4}$ lies in Quadrant IV, its $x$-coordinate is positive (x > 0) and its $y$-coordinate is negative (y < 0).

  • Final desired terminal point: $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

Terminal Points and Circular Motion

• Definition of Circumference and Terminal Point Motion:

  • A terminal point on the unit circle is the point reached after traveling a given distance $t$ along the circle, starting at $(1, 0)$.

  • Circumference formula for a circle of radius $r$:     $C = 2\pi r$

  • Substituting $r = 1$ gives total circumference:     $C = 2\pi$

• Counterclockwise Motion and Terminal Points:

  • Complete trip around the unit circle once corresponds to a distance of $2\pi$.

  • Starting Position ($t = 0$):

  • Terminal point: $(1, 0)$

  • Quarter trip around the circle ($t = \frac{1}{4} \times 2\pi = \frac{\pi}{2}$):

  • Terminal point: $(0, 1)$

  • Halfway trip around the circle ($t = \frac{1}{2} \times 2\pi = \pi$):

  • Terminal point: $(-1, 0)$

  • Three-quarters trip around the circle ($t = \frac{3}{4} \times 2\pi = \frac{3\pi}{2}$):

  • Terminal point: $(0, -1)$

  • Full trip around the circle ($t = 2\pi$):

  • Returns to terminal point $(1, 0)$

• Clockwise Motion (Negative Values of $t$):

  • Traveling clockwise (opposite direction) is indicated by a negative sign.

  • Quarter trip clockwise ($t = -\frac{\pi}{2}$):

  • Reaches terminal point $(0, -1)$

  • Half trip clockwise ($t = -\pi$):

  • Reaches terminal point $(-1, 0)$

• Equivalence of Multiple Terminal Points:

  • Different values and multiples of $\pi$ can lead to the exact same terminal points (for example, $t = \frac{3\pi}{2}$ and $t = -\frac{\pi}{2}$ both share the terminal point $(0, -1)$; $t = \pi$ and $t = -\pi$ both share $(-1, 0)$).

A unit circle is a circle with a radius of $1$ centered at the origin $(0, 0)$.

• It shows all points that are $1$ unit away from the center.

• The equation for the unit circle is:
  $x^2 + y^2 = 1$

• Important points on the unit circle are:

  • On the right: $(1, 0)$

  • On the top: $(0, 1)$

  • On the left: $(-1, 0)$

  • On the bottom: $(0, -1)$

• To find other points on the unit circle, you can use the equation and the quadrant where the point is located (four sections of the circle).

Reference Numbers and Quadrants

• A reference number helps you find where a point is on the circle based on its angle.

• Every angle can be simplified by finding its position in the first quadrant.

• Here’s how to use reference numbers:

  • 1st Quadrant: Both x and y are positive.

  • 2nd Quadrant: x is negative, y is positive.

  • 3rd Quadrant: Both x and y are negative.

  • 4th Quadrant: x is positive, y is negative.

Examples

• If a point in the third quadrant has $x = -\frac{\sqrt{3}}{2}$, here’s how to find $y$:

  1. Plug it into the circle’s equation:
     $\left(-\frac{\sqrt{3}}{2}\right)^2 + y^2 = 1$.

  2. Solve: $\frac{3}{4} + y^2 = 1$.

  3. This gives: $y^2 = \frac{1}{4}$, so $y = -\frac{1}{2}$ (because it’s in the third quadrant).

  4. Complete point: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$.

• If $t = \frac{5\pi}{6}$, it is in the second quadrant. Reference number: $\bar{t} = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$.