The unit circle is the circle with radius 1 centered at the origin, x2 + y2 = 1. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ). This lets sine and cosine work for every angle, including obtuse and negative angles, not just the acute angles of right triangles.
In Spanish
círculo unitario
Also called
trigonometric circle, unit circle chart, circle of radius 1
On a circle of radius 1, the right triangle under the radius has hypotenuse 1, so its legs are the cosine and the sine.
Formal definition and notation
The unit circle is {(x, y) : x2 + y2 = 1}. Place an angle θ in standard position, with its vertex at the origin and its initial side on the positive x-axis, turning counterclockwise for positive angles. Its terminal side meets the circle at (x, y), and by definition cos θ = x, sin θ = y and tan θ = yx for x ≠ 0. Because the radius is 1, the arc length from (1, 0) equals the angle in radians, and the equation of the circle gives cos2θ + sin2θ = 1.
A circle can have any center and radius, such as (x − 2)2 + y2 = 9. The unit circle is the one with center (0, 0) and radius exactly 1.
Unit Circlevs.Reference angle
A reference angle is the acute angle to the x-axis, such as 30° for 150°. It gives the size of the coordinates, while the unit circle point also carries the signs.
Quick check: 3 questions
Question 1 of 3
What are the coordinates of the point at 180 degrees on the unit circle?
Show answer and explanation
Answer: C. A half turn from (1, 0) lands on the left end: (−1, 0). (0, 1) is 90°, (0, −1) is 270°, and (1, 0) is 0°.
Question 2 of 3
What is sin 210°?
Show answer and explanation
Answer: B.210° is in Quadrant III with reference angle 30°, so y = −12. 12 misses the sign, and the √32 choices are cosine sizes.
Question 3 of 3
Which point lies on the unit circle?
Show answer and explanation
Answer: D.0.62 + (−0.8)2 = 0.36 + 0.64 = 1. The others give 0.5, 2 and 0.72, so they are not 1 unit from the origin.
Unit Circle: frequently asked questions
How do you memorize the unit circle?
Learn Quadrant I only: 0°, 30°, 45°, 60°, 90°, whose coordinates use only 0, 12, √22, √32 and 1. Every other angle copies one of these by its reference angle; then attach signs by quadrant. Remember that the x-coordinate is the cosine and the y-coordinate is the sine.
Why is the radius of the unit circle 1?
With radius 1, the hypotenuse of the right triangle under any radius is 1, so cos θ = x1 = x and sin θ = y1 = y. The coordinates of the point simply are the cosine and sine. A radius of 1 also makes the arc length equal to the angle in radians.
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