The Unit Circle Explorer
Sine and cosine as coordinates, reference angles, all six ratios
Step 1 of 11 -- Phase 1: Building from Simulation 7
Here is a right triangle inside a circle of radius 1. The hypotenuse IS the radius. What is the length of the hypotenuse?
Step 2 of 11
From Simulation 7, you know that \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\). The hypotenuse is 1. So what does sine simplify to?
Step 3 of 11
The opposite side is vertical -- it measures your height above or below the x-axis. What is another name for vertical position on a graph?
Step 4 of 11 -- Phase 2: Confirming Known Values
The point has moved to 30 degrees. In Simulation 7, you found \(\sin 30° = 0.5\). Read the y-coordinate of the point on the circle. What is it?
Step 5 of 11
Now check at 45 degrees and 60 degrees. At 45 degrees the y-coordinate reads about 0.707, and at 60 degrees it reads about 0.866. Do these match \(\sin 45°\) and \(\sin 60°\) from Simulation 7?
Step 6 of 11 -- Phase 3: Past 90 Degrees
The point is at 120 degrees. There is no right triangle with a 120-degree angle. So is sine even defined here?
Step 7 of 11
A vertical line from the point to the x-axis creates a reference triangle. Its acute angle is 60 degrees -- the same shape as the 60-degree triangle. What is different about the coordinates?
Step 8 of 11
The unit circle has extended sine and cosine to every angle. Now predict the sign of x and y in each quadrant. Click each quadrant to set your prediction, then reveal.
Quadrant I (0 to 90)
x: +, y: +
Quadrant II (90 to 180)
Quadrant III (180 to 270)
Quadrant IV (270 to 360)
Step 9 of 11 -- Phase 4: The Other Four Ratios
You have sine as y and cosine as x. Tangent is \(\frac{\text{opposite}}{\text{adjacent}}\), which on the unit circle is \(\frac{y}{x}\). What does that ratio represent geometrically?
Step 10 of 11
The reciprocal ratios follow: \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), \(\cot\theta = \frac{1}{\tan\theta}\). When \(\sin\theta = 0\), what happens to \(\csc\theta\)?
Step 11 of 11 -- Phase 5: Reading the Circle
Click on the unit circle: where is \(\sin\theta = 0\)?
Click on the circle at the angle where sine equals zero.
Explore the Unit Circle
Drag the point around the circle to see how coordinates and trig ratios change. Use arrow keys for precise control.
| Angle | 0 |
| Radians | 0 |
| Quadrant | I |
| Ref. Angle | 0 |
| sin | 0 |
| cos | 1 |
| tan | 0 |
| csc | -- |
| sec | 1 |
| cot | -- |
The point is at 0 degrees (0 radians), on the positive x-axis. Coordinates: (1, 0). Sine is the y-coordinate, cosine is the x-coordinate.
Try This
Ferris Wheel
A Ferris wheel has a 50-foot radius, and its center sits 55 feet above the ground. Your height at rotation angle \(\theta\) (measured from the 3 o'clock position) is:
Question 1: What is your height at \(\theta = 90°\)?
Enter your answer in feet (within 1 foot).
Question 2: What is your height at \(\theta = 270°\)?
Enter your answer in feet (within 1 foot).
Question 3: Does the answer make physical sense?
Clock Second Hand
A clock's second hand is 8 inches long, pinned at the origin. Where is the tip at 20 seconds past the hour?
Careful: clocks run clockwise and start at 12 (the top), but the unit circle runs counterclockwise and starts at 3 (the right). You need to convert.
At 20 seconds, the second hand has swept \(\frac{20}{60} = \frac{1}{3}\) of a full rotation, or 120 degrees clockwise from 12. Converting to unit-circle angle: start at 90 degrees (12 o'clock position) and subtract 120 degrees.
Question: What is the unit-circle angle for 20 seconds past the hour?
Follow-up: At \(330°\) on the unit circle (radius 8), the tip coordinates are \((8\cos 330°,\; 8\sin 330°)\). What are the approximate coordinates?
Radar Dish
A rotating radar dish sweeps a full circle. Its horizontal reach is proportional to \(|\cos\theta|\). For what range of angles is the horizontal reach more than 70% of maximum?
In other words, find all angles \(\theta\) where \(|\cos\theta| > 0.7\).
Question: Which ranges of angles satisfy \(|\cos\theta| > 0.7\)? Select all that apply.
Select all correct ranges, then check.
- Define sine and cosine as coordinates of a point on the unit circle
- Read exact and approximate trig values from the unit circle for any angle
- Use reference angles and quadrant signs to evaluate trig functions beyond 90 degrees
- Identify where each of the six trig functions is positive, negative, or undefined
- Apply unit-circle reasoning to real-world rotation problems
Quick Check
What is \(\cos 150°\)?
Instructor Notes
Teaching Notes
This is the central simulation in the trig sequence. Every later simulation builds on the identification of sine and cosine as coordinates. Spend time here. The discover sequence is deliberately longer (11 steps, 5 phases) because this one idea -- that trig functions are coordinates, not just ratios -- is worth the extra scaffolding.
Phase 1 (steps 1-3) connects back to Simulation 7 so students see that the unit circle is not replacing what they learned but reframing it. Phase 3 (steps 6-8) is where the real payoff comes: extending trig to angles past 90 degrees by reading coordinates instead of building triangles.
Common Student Errors
- Believing sine and cosine only apply to acute angles in right triangles. Steps 6-8 address this directly.
- Mixing up which coordinate is which: reading (x, y) as (sin, cos) instead of (cos, sin). The color coding (red for sine/y, blue for cosine/x) reinforces the correct mapping.
- Forgetting reference angles -- trying to memorize Q2/Q3/Q4 values as new facts instead of reading them from the triangle shape and coordinate signs.
- Confusion about tangent undefined at 90 and 270. Step 10 builds the intuition that division by zero produces "no answer," not infinity.
Discussion Questions
- Why does the unit circle use radius 1? What would change if the radius were 2?
- A student says "sine is always positive." What angles would you show them?
- How does the Ferris wheel problem connect to the unit circle? Where does the "+55" come from physically?
Exam Connection
Questions asking for exact trig values at multiples of 30 and 45 degrees are standard. Students should be able to determine the sign from the quadrant and the magnitude from the reference angle, not from memorization. The unit circle diagram is often allowed on exams, so teach students to read it, not recite it.