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Unit Circle



Definition and Equation
Coordinates as Trigonometric Values
Standard Position and the Terminal Side
The Four Quadrants and Sign Patterns
Reference Angles
Exact Values for the Standard Angles
The Pythagorean Identity
Angles Beyond One Rotation and Negative Angles
From Circular Motion to Waves



Extending Trigonometry Beyond the Right Triangle

A circle of radius 1, centered at the origin — nothing more. Yet this minimal construction carries the entire weight of extending trigonometric functions from acute-angle ratios to functions defined on all real numbers. The idea is direct: place an angle θ\theta in standard position, let its terminal side intersect the unit circle at a point PP, and define cos⁡θ\cos\theta and sin⁡θ\sin\theta as the xx- and yy-coordinates of PP. No triangle is needed. No restriction to acute angles applies. Negative angles, obtuse angles, angles spanning multiple rotations — all receive well-defined trigonometric values through this single geometric device.

The unit circle reveals structure that right triangle trigonometry cannot access. It organizes the signs of the trigonometric functions by quadrant, introduces reference angles as a computational shortcut, provides exact coordinates for the sixteen standard angles, and gives geometric meaning to periodicity — the fact that adding a full rotation returns to the same point and therefore the same function values. The Pythagorean identity cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1 is simply the equation of the circle itself, rewritten in trigonometric language. Every pattern visible on the graphs of sine and cosine, every symmetry catalogued among the properties, and every sign rule used in solving equations traces back to this circle.



Definition and Equation

The unit circle is defined by two conditions: its center is at the origin of the coordinate plane, and its radius is 1. Every point (x,y)(x, y) on the circle satisfies:

x2+y2=1x^2 + y^2 = 1


This equation is a direct application of the Pythagorean theorem (or equivalently, the distance formula): a point lies on the unit circle if and only if its distance from the origin is exactly 1.

The choice of radius 1 is deliberate. It eliminates the denominator from the right triangle definitions. In a right triangle inscribed in a circle of radius rr, the sine of an angle is oppositer\frac{\text{opposite}}{r} and the cosine is adjacentr\frac{\text{adjacent}}{r}. When r=1r = 1, these reduce to simply the opposite and adjacent sides — or equivalently, the yy- and xx-coordinates of the point where the terminal side meets the circle. The unit circle strips away the division and makes the coordinates themselves equal to the trigonometric values.

Any circle of radius rr could serve a similar purpose — the values would just need to be divided by rr. But with r=1r = 1, no division is necessary, and the connection between geometry and function values is as direct as possible.

Coordinates as Trigonometric Values

For any angle θ\theta placed in standard position (vertex at the origin, initial side along the positive xx-axis), the terminal side intersects the unit circle at exactly one point. The coordinates of that point define cosine and sine:

(cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta)


The xx-coordinate is cos⁡θ\cos\theta. The yy-coordinate is sin⁡θ\sin\theta. This is the definition — not a derived result, not an approximation, but the way sine and cosine are defined for arbitrary angles.

From these, the remaining four functions follow:

tan⁡θ=sin⁡θcos⁡θ=yx(x≠0)\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x} \quad (x \neq 0)


cot⁡θ=cos⁡θsin⁡θ=xy(y≠0)\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{x}{y} \quad (y \neq 0)


sec⁡θ=1cos⁡θ=1x(x≠0)\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x} \quad (x \neq 0)


csc⁡θ=1sin⁡θ=1y(y≠0)\csc\theta = \frac{1}{\sin\theta} = \frac{1}{y} \quad (y \neq 0)


For acute angles in the first quadrant, this definition agrees perfectly with the right triangle ratios. Dropping a perpendicular from the point P=(cos⁡θ,sin⁡θ)P = (\cos\theta, \sin\theta) to the xx-axis creates a right triangle with hypotenuse 1 (the radius), horizontal leg cos⁡θ\cos\theta, and vertical leg sin⁡θ\sin\theta. The ratios opposite/hypotenuse and adjacent/hypotenuse reproduce sin⁡θ\sin\theta and cos⁡θ\cos\theta exactly. The unit circle definition is a generalization that preserves everything from the triangle definition while removing the constraint that θ\theta be acute.

Standard Position and the Terminal Side

    An angle is in standard position when its vertex sits at the origin and its initial side lies along the positive xx-axis. The terminal side is the ray obtained by rotating the initial side through the angle θ\theta. The direction of rotation determines the sign: counterclockwise is positive, clockwise is negative.

    The terminal side is what matters for trigonometric evaluation. It determines where the angle "lands" on the unit circle and therefore what point (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) it produces. Two different rotations that end at the same terminal side — such as 45°45° and 405°405°, or 120°120° and −240°-240° — produce the same point on the circle and therefore the same values of all six trigonometric functions.

    Angles whose terminal sides fall along the coordinate axes — 0°0°, 90°90°, 180°180°, 270°270°, and their equivalents — are called quadrantal angles. Their trigonometric values come directly from the axis coordinates:

  • θ=0°\theta = 0°: point (1,0)(1, 0) → cos⁡(0°)=1\cos(0°) = 1, sin⁡(0°)=0\sin(0°) = 0
  • θ=90°\theta = 90°: point (0,1)(0, 1) → cos⁡(90°)=0\cos(90°) = 0, sin⁡(90°)=1\sin(90°) = 1
  • θ=180°\theta = 180°: point (−1,0)(-1, 0) → cos⁡(180°)=−1\cos(180°) = -1, sin⁡(180°)=0\sin(180°) = 0
  • θ=270°\theta = 270°: point (0,−1)(0, -1) → cos⁡(270°)=0\cos(270°) = 0, sin⁡(270°)=−1\sin(270°) = -1

    • At quadrantal angles, some trigonometric functions are undefined. Tangent and secant require cos⁡θ≠0\cos\theta \neq 0, so they are undefined at 90°90° and 270°270°. Cotangent and cosecant require sin⁡θ≠0\sin\theta \neq 0, so they are undefined at 0°0° and 180°180°. These points of undefinedness correspond to vertical asymptotes on the graphs and to excluded values in the domains of the trigonometric functions.

The Four Quadrants and Sign Patterns

The coordinate axes divide the plane into four quadrants, and the signs of xx and yy in each quadrant determine the signs of the trigonometric functions.

Quadrant I (0°<θ<90°0° < \theta < 90°): both x>0x > 0 and y>0y > 0. All six trigonometric functions are positive.

Quadrant II (90°<θ<180°90° < \theta < 180°): x<0x < 0, y>0y > 0. Sine and cosecant are positive (they depend on yy). Cosine, secant, tangent, and cotangent are negative.

Quadrant III (180°<θ<270°180° < \theta < 270°): both x<0x < 0 and y<0y < 0. Tangent and cotangent are positive (the ratio of two negatives is positive). Sine, cosine, cosecant, and secant are negative.

Quadrant IV (270°<θ<360°270° < \theta < 360°): x>0x > 0, y<0y < 0. Cosine and secant are positive (they depend on xx). Sine, cosecant, tangent, and cotangent are negative.

The mnemonic ASTC — "All, Sine, Tangent, Cosine" — lists which group of functions is positive in each quadrant, starting from Quadrant I and moving counterclockwise. Various phrase mnemonics exist ("All Students Take Calculus" being the most common), but the underlying logic is more reliable than any mnemonic: cos⁡θ\cos\theta has the same sign as xx, sin⁡θ\sin\theta has the same sign as yy, and tan⁡θ\tan\theta is positive when xx and yy share the same sign.

Knowing the quadrant of an angle immediately determines the sign of every trigonometric function at that angle. This information, combined with the magnitude obtained from a reference angle, is sufficient to evaluate any trigonometric function at any standard angle.

Reference Angles

    The reference angle for a given angle θ\theta is the acute angle formed between the terminal side of θ\theta and the xx-axis. It is always positive and always lies between 0°0° and 90°90° (or between 00 and π2\frac{\pi}{2}). The reference angle captures the "magnitude" component of the trigonometric evaluation, while the quadrant determines the sign.

    The computation depends on the quadrant:

  • Quadrant I: reference angle =θ= \theta
  • Quadrant II: reference angle =180°−θ= 180° - \theta (or π−θ\pi - \theta)
  • Quadrant III: reference angle =θ−180°= \theta - 180° (or θ−π\theta - \pi)
  • Quadrant IV: reference angle =360°−θ= 360° - \theta (or 2π−θ2\pi - \theta)

    • The key property: the trigonometric function values at any angle θ\theta equal the function values at the reference angle, up to a sign determined by the quadrant. Formally:

    ∣sin⁡θ∣=sin⁡(reference angle),∣cos⁡θ∣=cos⁡(reference angle)|\sin\theta| = \sin(\text{reference angle}), \quad |\cos\theta| = \cos(\text{reference angle})


    This reduces every trigonometric evaluation to two steps: (1) find the reference angle and evaluate the function there (using the known values from the special right triangles), and (2) attach the correct sign based on the quadrant.

    For example, to evaluate sin⁡(225°)\sin(225°): the angle is in Quadrant III, so the reference angle is 225°−180°=45°225° - 180° = 45°. The sine of 45°45° is 22\frac{\sqrt{2}}{2}. Sine is negative in Quadrant III, so sin⁡(225°)=−22\sin(225°) = -\frac{\sqrt{2}}{2}.

    To evaluate cos⁡(5π6)\cos\left(\frac{5\pi}{6}\right): the angle is in Quadrant II (between π2\frac{\pi}{2} and π\pi), so the reference angle is π−5π6=π6\pi - \frac{5\pi}{6} = \frac{\pi}{6}. The cosine of π6\frac{\pi}{6} is 32\frac{\sqrt{3}}{2}. Cosine is negative in Quadrant II, so cos⁡(5π6)=−32\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}.

    Reference angles work because the unit circle has reflective symmetry across both axes. Points in different quadrants that share the same reference angle are mirror images of each other, differing only in the signs of their coordinates.

Exact Values for the Standard Angles

    The sixteen standard angles — every multiple of 30°30° and 45°45° from 0°0° to 360°360° — each have exact trigonometric values expressible using only integers, fractions, and square roots. These values are not approximations; they are the precise coordinates of the corresponding points on the unit circle.

    The first-quadrant values come directly from the special right triangles:

  • (cos⁡0°,sin⁡0°)=(1,0)(\cos 0°, \sin 0°) = (1, 0)
  • (cos⁡30°,sin⁡30°)=(32,12)(\cos 30°, \sin 30°) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)
  • (cos⁡45°,sin⁡45°)=(22,22)(\cos 45°, \sin 45°) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
  • (cos⁡60°,sin⁡60°)=(12,32)(\cos 60°, \sin 60°) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)
  • (cos⁡90°,sin⁡90°)=(0,1)(\cos 90°, \sin 90°) = (0, 1)

    • Every other standard angle shares one of these reference angles — 30°30°, 45°45°, or 60°60° — and its coordinates are obtained by adjusting signs according to the quadrant. For instance, 150°150° has reference angle 30°30° and lies in Quadrant II, so cos⁡(150°)=−32\cos(150°) = -\frac{\sqrt{3}}{2} and sin⁡(150°)=12\sin(150°) = \frac{1}{2}. The angle 315°315° has reference angle 45°45° and lies in Quadrant IV, so cos⁡(315°)=22\cos(315°) = \frac{\sqrt{2}}{2} and sin⁡(315°)=−22\sin(315°) = -\frac{\sqrt{2}}{2}.

    A useful observation: only five magnitudes ever appear as coordinates on the unit circle at standard angles: 00, 12\frac{1}{2}, 22\frac{\sqrt{2}}{2}, 32\frac{\sqrt{3}}{2}, and 11. The entire unit circle diagram, with all sixteen labeled points, is built from just these five values with appropriate sign changes. Recognizing this pattern — rather than memorizing sixteen separate pairs — is the efficient path to fluency.

    The sine values at 0°,30°,45°,60°,90°0°, 30°, 45°, 60°, 90° follow an ascending pattern: 02,12,22,32,42\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2} — that is, 0,12,22,32,10, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1. Cosine follows the same values in reverse order. This mnemonic, while not a proof, provides a quick reconstruction of the first-quadrant values.

The Pythagorean Identity

Every point on the unit circle satisfies x2+y2=1x^2 + y^2 = 1. Since x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta, substitution gives:

cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1


This is the Pythagorean identity — the most fundamental equation in trigonometry. It holds for every real number θ\theta, without exception. It is not an approximation, not a special case, and not restricted to any particular quadrant or angle type. It is an algebraic restatement of the geometric fact that sine and cosine are coordinates on a circle of radius 1.

The identity has two standard rearrangements:

sin⁡2θ=1−cos⁡2θandcos⁡2θ=1−sin⁡2θ\sin^2\theta = 1 - \cos^2\theta \qquad \text{and} \qquad \cos^2\theta = 1 - \sin^2\theta


These substitutions appear constantly in simplification and equation solving. If an expression contains sin⁡2θ\sin^2\theta and it would be simpler in terms of cos⁡θ\cos\theta, replace sin⁡2θ\sin^2\theta with 1−cos⁡2θ1 - \cos^2\theta. The reverse substitution is equally common.

Dividing the Pythagorean identity by cos⁡2θ\cos^2\theta (where cos⁡θ≠0\cos\theta \neq 0) produces:

1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta


Dividing by sin⁡2θ\sin^2\theta (where sin⁡θ≠0\sin\theta \neq 0) produces:

1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta


These are the three Pythagorean identities — all consequences of a single circle equation. They are among the most frequently used tools in the entire subject, appearing in identity proofs, expression simplification, equation solving, integration, and the derivation of other formulas.

Angles Beyond One Rotation and Negative Angles

The unit circle imposes no upper or lower bound on angles. An angle of 400°400° means a full counterclockwise rotation (360°360°) followed by an additional 40°40° — the terminal side lands at the same position as 40°40°. An angle of −150°-150° means a clockwise rotation of 150°150°. An angle of 1000°1000° means two full rotations (720°720°) plus 280°280°, so the terminal side coincides with that of 280°280°.

This is the geometric meaning of coterminal angles: different amounts of rotation that end at the same point on the circle. Since the trigonometric values depend only on the terminal point — not on how many times the terminal side wound around to get there — coterminal angles always produce identical function values. The angle θ\theta, the angle θ+360°\theta + 360°, the angle θ−360°\theta - 360°, and the angle θ+720°\theta + 720° all yield the same sine, the same cosine, and the same everything else.

This wrapping behavior is the source of periodicity. The statement sin⁡(θ+2π)=sin⁡θ\sin(\theta + 2\pi) = \sin\theta is not an algebraic identity to be proved — it is a geometric inevitability. After a full rotation of 2π2\pi radians, the point on the circle returns to its starting position, and the yy-coordinate (sine) is unchanged. The same holds for cosine and for all functions derived from them. Tangent and cotangent repeat after just π\pi radians because the ratio yx\frac{y}{x} returns to its original value after a half rotation (both xx and yy reverse sign, and the ratio of two negatives equals the ratio of the original positives).

The ability to handle arbitrary angles — large, small, negative, irrational multiples of π\pi — is what transforms the trigonometric ratios of right triangle trigonometry into genuine functions of a real variable, with the full apparatus of analysis (continuity, differentiability, integrability) available.

From Circular Motion to Waves

Imagine a point PP moving counterclockwise around the unit circle at a constant rate. At any moment, the angle θ\theta swept from the positive xx-axis determines the position P=(cos⁡θ,sin⁡θ)P = (\cos\theta, \sin\theta). Now track just the yy-coordinate as θ\theta increases. It starts at 00 (when θ=0\theta = 0), rises to 11 (at θ=π2\theta = \frac{\pi}{2}), returns to 00 (at θ=π\theta = \pi), drops to −1-1 (at θ=3π2\theta = \frac{3\pi}{2}), and returns to 00 (at θ=2π\theta = 2\pi). Then the pattern repeats.

Plot this yy-coordinate against θ\theta, and the result is the familiar wave shape of the sine graph. The xx-coordinate, tracked similarly, produces the cosine graph — the same wave, shifted by π2\frac{\pi}{2}.

This connection between circular motion and wave motion is not merely a visual analogy. It is the mathematical reason why trigonometric functions model oscillations. Any quantity that varies by cycling through a fixed pattern — a pendulum swinging, a voltage alternating, a season repeating — follows a path that can be described (or closely approximated) by sines and cosines. The unit circle is the generator, and the wave is its projection.

This perspective also explains why the graphs of sine and cosine are smooth, continuous, and bounded between −1-1 and 11: coordinates on a unit circle must lie in [−1,1][-1, 1], and a point moving at constant speed around a smooth curve traces a smooth function. The amplitude corresponds to the radius (which is 1 for the unit circle), and the period corresponds to the time for one full revolution (2π2\pi radians).