Unit Circle Values Reference
Review common degree and radian angles with sine, cosine, and tangent values. The unit circle connects angle measure to the coordinate pair where cosine is x and sine is y.
Reference table
8 angles
Common first-quadrant and axis angles for fast classroom review.
Common degree, radian, sine, cosine, and tangent values
| Degrees | Radians | cos(theta) | sin(theta) | tan(theta) |
|---|---|---|---|---|
| 0 deg | 0 | 1 | 0 | 0 |
| 30 deg | pi/6 | sqrt(3)/2 | 1/2 | sqrt(3)/3 |
| 45 deg | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 deg | pi/3 | 1/2 | sqrt(3)/2 | sqrt(3) |
| 90 deg | pi/2 | 0 | 1 | undefined |
| 180 deg | pi | -1 | 0 | 0 |
| 270 deg | 3pi/2 | 0 | -1 | undefined |
| 360 deg | 2pi | 1 | 0 | 0 |
Cosine is x, sine is y
On the unit circle, each angle points to a coordinate pair on a circle with radius 1. The x-coordinate is cosine and the y-coordinate is sine. Tangent is sine divided by cosine, so tangent is undefined when cosine equals zero.
Degree values are often easier to visualize, while radian values connect directly to arc length and advanced trigonometry. Use the converter when you need the same angle in both systems.
Read every standard angle from one coordinate rule
A unit-circle point is not a list to memorize in isolation. At an angle theta, the point is (cos theta, sin theta). The first-quadrant magnitudes repeat through the other quadrants, while the x- and y-signs change with the point's location.
Sign pattern by quadrant
Angles start on the positive x-axis and increase counterclockwise. Tangent follows sin/cos, so its sign comes from the sine and cosine signs.
- Quadrant I
- cosine positive, sine positive, tangent positive.
- Quadrant II
- cosine negative, sine positive, tangent negative.
- Quadrant III
- cosine negative, sine negative, tangent positive.
- Quadrant IV
- cosine positive, sine negative, tangent negative.
- Horizontal axis
- sine is zero, so tangent is zero.
- Vertical axis
- cosine is zero, so tangent is undefined.
Use a reference angle for magnitude
A reference angle is the acute angle between the terminal side and the x-axis. Standard angles such as 30, 45, and 60 degrees provide the same absolute coordinate values in every quadrant.
For 150 degrees, the reference angle is 30 degrees. The magnitude pair is therefore (sqrt(3)/2, 1/2), and quadrant II changes only the x-sign: (-sqrt(3)/2, 1/2).
Connect degrees, radians, and turns
A half turn is 180 degrees or pi radians, and a quarter turn is 90 degrees or pi/2 radians. Adding a full 2pi-radian turn returns to the same coordinate.
Use radian labels when working with arc length, calculus, or identities. Use degrees when a diagram or instruction expresses familiar portions of a turn. The coordinate does not depend on which label system names the angle.
Interpret tangent as a ratio
Tangent equals sine divided by cosine, or y/x on the unit circle. At 45 degrees both coordinates have the same nonzero magnitude, so the ratio is 1.
At 90 and 270 degrees, x equals zero. Division by zero is undefined, which is why the reference table does not show a finite tangent value for the vertical-axis angles.
Worked check: locate 150 degrees
- 1Find the reference angle: 180 - 150 = 30 degrees.
- 2Start with the 30-degree magnitudes cos = sqrt(3)/2 and sin = 1/2.
- 3Apply quadrant II signs to get (-sqrt(3)/2, 1/2); tangent is (1/2)/(-sqrt(3)/2) = -sqrt(3)/3.
Reference boundaries
- The visible table highlights common axis and first-quadrant angles. Apply reference-angle and sign rules for standard angles such as 120, 135, 150, 210, 225, 240, 300, 315, and 330 degrees.
- Exact radicals are preferable in symbolic work. Decimal values can be useful for measurement or software inputs, but their precision depends on the rounding you choose.
- The unit circle defines trigonometric values for angles; it does not by itself solve a triangle. A triangle problem also needs side, angle, and geometry constraints.
Sources and definitions
- NIST SP 330, Section 5: angle units
Reference for degree and radian conventions used in the table and diagram.
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