The unit circle is a circle with radius 1 centered at the origin of the x-y coordinate plane. It is one of the most useful diagrams in geometry and trigonometry because it turns angles into coordinates. Every point on the circle has coordinates that connect directly to cosine and sine.
This makes the unit circle a visual tool for understanding periodic motion, waves, rotations, and triangles.
An angle in standard position starts on the positive x-axis and rotates around the origin. Where its terminal ray meets the unit circle, the x-coordinate is cos θ and the y-coordinate is sin θ. Special angles such as 30°, 45°, 60°, and 90° have exact coordinate values that appear often in math and science.
By using symmetry across the four quadrants, you can find sine and cosine values for many angles without memorizing every one separately.
Key Facts
- The unit circle is defined by x^2 + y^2 = 1.
- For an angle θ on the unit circle, the point is (cos θ, sin θ).
- cos θ is the x-coordinate and sin θ is the y-coordinate.
- Angles measured counterclockwise from the positive x-axis are positive.
- One full rotation is 360° = 2π radians.
- tan θ = sin θ / cos θ, when cos θ is not 0.
Vocabulary
- Unit circle
- A circle with radius 1 centered at the origin of a coordinate plane.
- Standard position
- An angle position where the vertex is at the origin and the initial side lies on the positive x-axis.
- Terminal ray
- The ray that shows where an angle ends after rotating from its initial side.
- Radian
- A unit of angle measure based on arc length, where 2π radians equals one full circle.
- Quadrant
- One of the four regions of the coordinate plane formed by the x-axis and y-axis.
Common Mistakes to Avoid
- Switching sine and cosine, which is wrong because cosine is the x-coordinate and sine is the y-coordinate on the unit circle.
- Forgetting quadrant signs, which gives incorrect values because coordinates can be positive or negative depending on the quadrant.
- Mixing degrees and radians, which is wrong because 60° and 60 radians represent very different rotations.
- Using tan θ when cos θ = 0, which is undefined because tan θ = sin θ / cos θ would require division by zero.
Practice Questions
- 1 Find the coordinates on the unit circle for θ = 60°. Then state sin 60° and cos 60°.
- 2 Convert 150° to radians, then use the unit circle to find sin 150° and cos 150°.
- 3 Explain why the points for 30° and 150° have the same sine value but different cosine values.