The Unit Circle: A Visual Guide to Sine and Cosine

Coordinates on the unit circle: (cos theta, sin theta)

The unit circle is a circle with radius exactly 1, centered at the origin (0,0) of a coordinate plane. For any angle theta, measured counterclockwise starting from the positive x-axis, drawing a line from the origin at that angle and finding where it crosses the circle gives a point whose x-coordinate equals cos(theta) and whose y-coordinate equals sin(theta) -- directly, with no scaling required, because the circle's radius is 1.

This is the geometric definition underlying every trigonometric identity: cosine is, by definition, the x-coordinate on the unit circle at a given angle, and sine is the y-coordinate. Tangent follows directly as the ratio of the two: tan(theta) = sin(theta) / cos(theta), which geometrically represents the slope of the line from the origin to that point on the circle.

Quadrant signs: where sine and cosine are positive or negative

The coordinate plane is divided into four quadrants by the x- and y-axes, and because sine and cosine are just the y- and x-coordinates of a point on the unit circle, their signs in each quadrant follow directly from the sign of coordinates in that quadrant.

This pattern is sometimes remembered with the mnemonic "All Students Take Calculus," referring to which functions are positive in each quadrant moving counterclockwise from quadrant I: All three functions positive in quadrant I, only Sine positive in quadrant II, only Tangent positive in quadrant III, and only Cosine positive in quadrant IV.

Reference angles: finding exact values outside quadrant I

A reference angle is the acute angle, always between 0 and 90 degrees, formed between an angle's terminal side and the nearest point on the x-axis. Reference angles make it possible to find the exact sine, cosine or tangent of any angle by relating it back to one of the five standard quadrant-I angles, then applying the correct sign for whichever quadrant the original angle actually falls in.

For example, an angle of 150 degrees falls in quadrant II, and its reference angle is 180 - 150 = 30 degrees. Since sine is positive in quadrant II, sin(150 degrees) equals positive sin(30 degrees) = 0.5. Cosine is negative in quadrant II, so cos(150 degrees) equals negative cos(30 degrees), approximately -0.8660. The reference angle supplies the magnitude; the quadrant supplies the sign.

The standard 0/30/45/60/90-degree table

Five angles -- 0, 30, 45, 60 and 90 degrees, all within quadrant I -- have sine and cosine values expressible as exact numbers rather than long decimals, and they form the foundation for finding exact trigonometric values at every other standard angle via reference angles. The table below lists both the exact form and the decimal approximation for each.

Notice the symmetry: the sine values from 0 to 90 degrees (0, 1/2, sqrt(2)/2, sqrt(3)/2, 1) are exactly the cosine values in reverse order, a direct consequence of sine and cosine being complementary functions -- sin(theta) = cos(90 degrees - theta) for any angle theta.

Coterminal angles and going beyond one rotation

Angles greater than 360 degrees, or negative angles, still correspond to a specific point on the unit circle -- they simply wrap around the circle one or more extra times. A coterminal angle is the equivalent angle within a single 0-to-360-degree rotation, found by adding or subtracting multiples of 360 degrees until the result falls in that range.

For example, an angle of 405 degrees has a coterminal angle of 405 - 360 = 45 degrees, meaning both angles point to the exact same location on the unit circle and therefore share identical sine, cosine and tangent values. Likewise, an angle of -30 degrees has a coterminal angle of -30 + 360 = 330 degrees, which falls in quadrant IV.

Câu hỏi thường gặp

Tài liệu tham khảo

1. Weisstein, Eric W. "Unit Circle" and "Reference Angle." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
2. Larson R. Trigonometry. Cengage Learning (unit circle definitions and the standard angle table).
3. Stewart J, Redlin L, Watson S. Precalculus: Mathematics for Calculus. Cengage Learning (unit circle and reference angles).