Key takeaway: The unit circle has radius 1 and sits at the origin; rotating a ray by θ away from the positive x-axis marks the point (cos θ, sin θ) on it.
Grade 11 – Algebra 2F-TF.A.2Trigonometry
Definition of Unit Circle
The unit circle is the circle \(x^2+y^2=1\): center at the origin, radius exactly 1. Start at the point \((1,0)\) and turn counterclockwise through an angle \(\theta\). The point where you stop has an x-coordinate equal to \(\cos\theta\) and a y-coordinate equal to \(\sin\theta\).
Because the radius is 1, no division is needed. That is why this circle extends sine and cosine to every angle, including angles bigger than 90\(^{\circ}\) and negative angles. The signs of the coordinates reveal the quadrant, and the tangent is the ratio \(\frac{y}{x}\).
Formula and notation
Rotate a ray by \(\theta\) from the positive x-axis, and the ray meets the circle at
\[(x,y)=(\cos\theta,\ \sin\theta), \qquad \tan\theta=\frac{y}{x}\ (x\neq 0).\]
Angles come in degrees or radians; one lap equals \(360^{\circ}\) or \(2\pi\), and counterclockwise is the positive direction.
Key facts
- Every point on the circle satisfies x² + y² = 1, which is the Pythagorean identity.
- Cosine is the horizontal coordinate and sine is the vertical coordinate.
- The four axis crossings correspond to quarter turns: 0°, 90°, 180° and 270°.
- Angles that are mirror images across an axis share absolute values and differ only in signs.
Unit Circle example
Use the unit circle to find the cosine, sine and tangent of \(300^{\circ}\).
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Since 300° is 60° short of a full turn, the ray lies in quadrant IV, making a 60° angle with the positive x-axis.
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A 60° ray reaches one half across and about 0.866 (that is, root three over two) up or down. In quadrant IV the horizontal part is positive and the vertical part is negative.
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Divide the vertical part by the horizontal part for the tangent: the halves cancel, leaving minus root three.
\(\cos 300^{\circ}=\frac{1}{2}\), \(\sin 300^{\circ}=-\frac{\sqrt{3}}{2}\), \(\tan 300^{\circ}=\)\(-\sqrt{3}\).
Common mistake
Watch out
Dario reads the point for 300° and says the sine is one half, since one half is the first number he sees.
Why it is wrong
He swapped the coordinates. The first number is the cosine; the sine is the second one, \(-\frac{\sqrt{3}}{2}\). A quick check on the unit circle: the ray at 300° is steep, so its vertical size is the larger one.
Related terms and lessons
- Reference Angle
- Coterminal Angles
- Pythagorean Identity
- Trigonometric Identities
- Inverse Trigonometric Functions
- Algebra 2 · Radians and the Unit CircleLessonWorksheetQuiz
- Precalculus · Angles, Radians and the Unit CircleLessonWorksheetQuiz
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