1. What is the Unit Circle?

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. It's fundamental in trigonometry, connecting angles to sine and cosine values.

2. How Does the Calculator Work?

The calculator uses the unit circle equation:

Where:

• \( x = \cos(\theta) \) — x-coordinate on the unit circle
• \( y = \sin(\theta) \) — y-coordinate on the unit circle
• \( \theta \) — Angle in degrees

Explanation: For any angle θ, the corresponding point on the unit circle has coordinates (cosθ, sinθ).

3. Importance of Unit Circle

Details: The unit circle is essential for understanding trigonometric functions, their periodic nature, and their relationships. It's used in mathematics, physics, engineering, and computer graphics.

4. Using the Calculator

Tips: Enter an angle in degrees (0-360) to see its corresponding point on the unit circle. The graph will display the angle and coordinates.

5. Frequently Asked Questions (FAQ)

Q1: Why is the radius 1?
A: A radius of 1 simplifies calculations, as the hypotenuse of the right triangle formed is always 1, making sine and cosine values directly equal to the y and x coordinates.

Q2: How are radians related?
A: Radians are another angle measurement where 360° = 2π radians. The calculator converts degrees to radians internally for calculations.

Q3: What are the key angles to remember?
A: 0° (1,0), 30° (√3/2,1/2), 45° (√2/2,√2/2), 60° (1/2,√3/2), 90° (0,1), etc.

Q4: Can I see multiple angles?
A: This calculator shows one angle at a time for clarity, but the graph displays the full unit circle with your selected angle highlighted.

Q5: How is this used in real applications?
A: The unit circle is fundamental in signal processing, circular motion analysis, and anywhere periodic functions are needed.