Unit Circle Calculator

How to Calculate Unit Circle Values

The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the coordinate plane. It's a fundamental concept in trigonometry that helps us understand the relationships between angles and their sine, cosine, and tangent values.

Formulas

For any point (x, y) on the unit circle corresponding to an angle θ:

• x-coordinate: \( x = \cos(\theta) \)
• y-coordinate: \( y = \sin(\theta) \)
• Tangent: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} \)

Where:

• θ is the angle in radians, measured counterclockwise from the positive x-axis
• x is the cosine of the angle and the x-coordinate on the unit circle
• y is the sine of the angle and the y-coordinate on the unit circle

Calculation Steps

1. Ensure the angle is in radians. If it's in degrees, convert it to radians by multiplying by π/180°.
2. Calculate the cosine of the angle to find the x-coordinate.
3. Calculate the sine of the angle to find the y-coordinate.
4. Calculate the tangent by dividing the sine by the cosine.

Example

Let's calculate the unit circle values for 45°:

1. Convert 45° to radians: 45° × (π/180°) = π/4 radians
2. Calculate x (cosine): cos(π/4) ≈ 0.7071
3. Calculate y (sine): sin(π/4) ≈ 0.7071
4. Calculate tangent: tan(π/4) = sin(π/4) / cos(π/4) = 1

Therefore, for an angle of 45°:

• Coordinates: (0.7071, 0.7071)
• Sine: 0.7071
• Cosine: 0.7071
• Tangent: 1

Visual Representation

This diagram illustrates the unit circle values for a 45° angle. The blue arc represents the angle, and the red line shows the radius to the point (0.7071, 0.7071) on the unit circle.