Unit Circle... What? Don't Panic!

If you looked at the board today and felt a moment of confusion, you aren't alone! The Unit Circle can look intimidating at first glance, but today we are going to demystify it. We are taking the concepts we learned yesterday about Special Right Triangles and simply placing them inside a circle.

The Big Idea

The "Unit" in Unit Circle simply means the radius is 1 ($r=1$). This simple fact changes everything because of our fundamental trigonometric definitions:

  • $\cos(\theta) = \frac{x}{r}$ becomes $\cos(\theta) = x$
  • $\sin(\theta) = \frac{y}{r}$ becomes $\sin(\theta) = y$

This means that every coordinate point $(x, y)$ on the circle actually represents $(\cos \theta, \sin \theta)$. If you know the coordinate, you know the trig values!

Connecting the Triangles

In the class notes attached below, you will see how we derived these values using the Pythagorean Theorem on our two favorite triangles:

  1. The $45^\circ-45^\circ-90^\circ$ Triangle: This gives us coordinates involving $\frac{\sqrt{2}}{2}$.
  2. The $30^\circ-60^\circ-90^\circ$ Triangle: This gives us coordinates involving $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.

Watch Your Signs (ASTC)

As we move around the circle from Quadrant I to Quadrant IV, the numbers stay the same, but the signs change. In our notes, we marked which functions are positive in which quadrants:

  • Quadrant I: All Positive
  • Quadrant II: Sine is Positive (so $y$ is positive)
  • Quadrant III: Tangent is Positive (since $x$ and $y$ are both negative)
  • Quadrant IV: Cosine is Positive (so $x$ is positive)

Today's Assignment

Your goal is to get comfortable with these positions. We are focusing strictly on degrees for now.

1. Fill in the Blank Unit Circle:
Use the attached Class Notes to fill in the "Blank Unit Circle" PDF. This is your study guide. Physically writing in the angles ($30^\circ, 45^\circ, 60^\circ...$) and their corresponding coordinates will help you memorize them.

2. Homework Worksheet:
Download the worksheet "Exact Trig Values of Special Angles."
Task: Complete all problems that are written in degrees (e.g., $\tan 60^\circ$, $\cos 135^\circ$). You can skip the radian problems (those with $\pi$) for now.

Check the attached files below for the notes, the blank template, and the worksheet. Good luck!