Unit Circle

Key Terms

Definition and Equation

Coordinates as Trigonometric Values

Standard Position and the Terminal Side

The Four Quadrants and Sign Patterns

Unit Circle Notation

Reference Angles

Exact Values for the Standard Angles

The Pythagorean Identity

Angles Beyond One Rotation and Negative Angles

From Circular Motion to Waves

Summary: The Sixteen Standard Angles

Unit Circle FAQ

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

\theta

in standard position, let its terminal side intersect the unit circle at a point

P

, and define

\cos\theta

and

\sin\theta

as the

x

- and

y

-coordinates of

P

. 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^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.

Key Terms

The Circle

Unit Circle— the circle

x^2 + y^2 = 1

centered at the origin

Sine—

y

-coordinate of the point at angle

\theta

Cosine—

x

-coordinate of the point at angle

\theta

Angle Placement

Standard Position— vertex at origin, initial side on positive

x

-axis

Initial Side— the fixed starting ray along the positive

x

-axis

Terminal Side— the ray after rotation, intersecting the circle

Angle Classification

Coterminal Angles— different rotations landing at the same point

Quadrantal Angles— terminal side on a coordinate axis

Reference Angle— acute angle to the

x

-axis, isolating magnitude from sign

Complementary Angles— two angles summing to

90°

Supplementary Angles— two angles summing to

180°

Formulas Used

Pythagorean Identity - Sine Cosine—

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

Pythagorean Identity - Tangent Secant—

1 + \tan^2\theta = \sec^2\theta

Pythagorean Identity - Cotangent Cosecant—

1 + \cot^2\theta = \csc^2\theta

See All Trigonometry Definitions →

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)

on the circle satisfies:

x^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

r

, the sine of an angle is

\frac{\text{opposite}}{r}

and the cosine is

\frac{\text{adjacent}}{r}

. When

r = 1

, these reduce to simply the opposite and adjacent sides — or equivalently, the

y

- and

x

-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

r

could serve a similar purpose — the values would just need to be divided by

r

. But with

r = 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

\theta

placed in standard position (vertex at the origin, initial side along the positive

x

-axis), the terminal side intersects the unit circle at exactly one point. The coordinates of that point define cosine and sine:

(\cos\theta, \sin\theta)

The

x

-coordinate is

\cos\theta

. The

y

-coordinate is

\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\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x} \quad (x \neq 0)

\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{x}{y} \quad (y \neq 0)

\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x} \quad (x \neq 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\theta, \sin\theta)

to the

x

-axis creates a right triangle with hypotenuse 1 (the radius), horizontal leg

\cos\theta

, and vertical leg

\sin\theta

. The ratios opposite/hypotenuse and adjacent/hypotenuse reproduce

\sin\theta

and

\cos\theta

exactly. The unit circle definition is a generalization that preserves everything from the triangle definition while removing the constraint that

\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

x

-axis. The terminal side is the ray obtained by rotating the initial side through the angle

\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\theta, \sin\theta)

it produces. Two different rotations that end at the same terminal side — such as

45°

and

405°

, or

120°

and

-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°

90°

180°

270°

, and their equivalents — are called quadrantal angles. Their trigonometric values come directly from the axis coordinates:

$\theta = 0°$ : point $(1, 0)$ → $\cos(0°) = 1$ , $\sin(0°) = 0$
$\theta = 90°$ : point $(0, 1)$ → $\cos(90°) = 0$ , $\sin(90°) = 1$
$\theta = 180°$ : point $(-1, 0)$ → $\cos(180°) = -1$ , $\sin(180°) = 0$
$\theta = 270°$ : point $(0, -1)$ → $\cos(270°) = 0$ , $\sin(270°) = -1$

At quadrantal angles, some trigonometric functions are undefined. Tangent and secant require

\cos\theta \neq 0

, so they are undefined at

90°

and

270°

. Cotangent and cosecant require

\sin\theta \neq 0

, so they are undefined at

0°

and

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

x

and

y

in each quadrant determine the signs of the trigonometric functions.

Quadrant I (

0° < \theta < 90°

): both

x > 0

and

y > 0

. All six trigonometric functions are positive.

Quadrant II (

90° < \theta < 180°

x < 0

y > 0

. Sine and cosecant are positive (they depend on

y

). Cosine, secant, tangent, and cotangent are negative.

Quadrant III (

180° < \theta < 270°

): both

x < 0

and

y < 0

. Tangent and cotangent are positive (the ratio of two negatives is positive). Sine, cosine, cosecant, and secant are negative.

Quadrant IV (

270° < \theta < 360°

x > 0

y < 0

. Cosine and secant are positive (they depend on

x

). 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\theta

has the same sign as

x

\sin\theta

has the same sign as

y

, and

\tan\theta

is positive when

x

and

y

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.

The sign distribution across all four quadrants, with the ASTC letter for each, collects into the table below.

Quadrant	Sign of x (cos θ)	Sign of y (sin θ)	Positive functions	ASTC
I (0°–90°)	+	+	all six	A
II (90°–180°)	−	+	sin, csc	S
III (180°–270°)	−	−	tan, cot	T
IV (270°–360°)	+	−	cos, sec	C

Unit Circle Notation

Notation

Unit Circle Notation

The circle's own marks: the coordinate pair that defines the functions, the Roman numerals the sign rules hang on, and the full-turn idiom behind every coterminal statement.

\sin

\cos

\tan

come from the right triangle page,

\theta

°

and bare-number radians from angle measure; the squared shorthand

\sin^2\theta

of The Pythagorean Identity below is taken up with the function pages.

(\cos\theta, \sin\theta)

the point cosine theta, sine theta

A coordinate pair doing definitional work: the terminal side meets the circle at exactly this point, and the coordinates are the function values — Coordinates as Trigonometric Values above. Cosine sits first because it is the

x

-coordinate.

CasesNamed points abbreviate to

P = (\cos\theta, \sin\theta)

; on a circle of radius

r

the pair scales to

(r\cos\theta, r\sin\theta)

— the bridge to polar description.

Do not confuseOrder. Sine feels first — alphabet, SOH-CAH-TOA habit — but is the second coordinate; reading

(\sin\theta, \cos\theta)

into the pair flips every point across the diagonal.

\text{I, II, III, IV}

quadrants one through four

Roman numerals, counterclockwise from the upper right — the labels every sign rule hangs on, as The Four Quadrants and Sign Patterns above lays out; working notes abbreviate to QI–QIV.

CasesASTC compresses the whole sign table into four letters — All, Sine, Tangent, Cosine positive, one per quadrant in order; the mnemonic phrases vary, the letter sequence never does.

Do not confuseReading order. Quadrant II sits to the left of I — the numerals follow the positive rotation direction, not left-to-right reading; misreading the direction swaps II with IV.

\theta + 360°

\theta + 2\pi

theta plus a full turn

Adding a full turn changes the label, not the point:

45°

and

405°

name one position — the pairs of Standard Position and the Terminal Side above, the periodicity

\sin(\theta + 2\pi) = \sin\theta

of Angles Beyond One Rotation below.

CasesSolving equations upgrades the idiom to a parameter — all coterminal labels at once, indexed by an integer

n

; the same wrap runs the complex exponential's periodic exponent.

Do not confuseEqual angles. As rotations,

45°

and

405°

are different motions — they collapse only as positions, and therefore as inputs to the trigonometric functions.

All trigonometry symbols →↑ Trigonometry

Reference Angles

The reference angle for a given angle

\theta

is the acute angle formed between the terminal side of

\theta

and the

x

-axis. It is always positive and always lies between

0°

and

90°

(or between

0

and

\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 $= \theta$
Quadrant II: reference angle $= 180° - \theta$ (or $\pi - \theta$ )
Quadrant III: reference angle $= \theta - 180°$ (or $\theta - \pi$ )
Quadrant IV: reference angle $= 360° - \theta$ (or $2\pi - \theta$ )

The key property: the trigonometric function values at any angle

\theta

equal the function values at the reference angle, up to a sign determined by the quadrant. Formally:

|\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°)

: the angle is in Quadrant III, so the reference angle is

225° - 180° = 45°

. The sine of

45°

\frac{\sqrt{2}}{2}

. Sine is negative in Quadrant III, so

\sin(225°) = -\frac{\sqrt{2}}{2}

To evaluate

\cos\left(\frac{5\pi}{6}\right)

: the angle is in Quadrant II (between

\frac{\pi}{2}

and

\pi

), so the reference angle is

\pi - \frac{5\pi}{6} = \frac{\pi}{6}

. The cosine of

\frac{\pi}{6}

\frac{\sqrt{3}}{2}

. Cosine is negative in Quadrant II, so

\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.

The four reference-angle formulas — one per quadrant, in both degree and radian forms — collect into a compact lookup table.

Quadrant of θ	Reference angle (degrees)	Reference angle (radians)
I	θ	θ
II	180° − θ	π − θ
III	θ − 180°	θ − π
IV	360° − θ	2π − θ

Exact Values for the Standard Angles

The sixteen standard angles — every multiple of

30°

and

45°

from

0°

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:

$(\cos 0°, \sin 0°) = (1, 0)$
$(\cos 30°, \sin 30°) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
$(\cos 45°, \sin 45°) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
$(\cos 60°, \sin 60°) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
$(\cos 90°, \sin 90°) = (0, 1)$

Every other standard angle shares one of these reference angles —

30°

45°

, or

60°

— and its coordinates are obtained by adjusting signs according to the quadrant. For instance,

150°

has reference angle

30°

and lies in Quadrant II, so

\cos(150°) = -\frac{\sqrt{3}}{2}

and

\sin(150°) = \frac{1}{2}

. The angle

315°

has reference angle

45°

and lies in Quadrant IV, so

\cos(315°) = \frac{\sqrt{2}}{2}

and

\sin(315°) = -\frac{\sqrt{2}}{2}

A useful observation: only five magnitudes ever appear as coordinates on the unit circle at standard angles:

0

\frac{1}{2}

\frac{\sqrt{2}}{2}

\frac{\sqrt{3}}{2}

, and

1

. 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°

follow an ascending pattern:

\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}

— that is,

0, \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

x^2 + y^2 = 1

. Since

x = \cos\theta

and

y = \sin\theta

, substitution gives:

Pythagorean Identity - Sine Cosine

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

Learn more about this formula: Pythagorean Identity - Sine Cosine →

Browse all trigonometry formulas →

This is the Pythagorean identity — the most fundamental equation in trigonometry. It holds for every real number

\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^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^2\theta

and it would be simpler in terms of

\cos\theta

, replace

\sin^2\theta

with

1 - \cos^2\theta

. The reverse substitution is equally common.

Dividing the Pythagorean identity by

\cos^2\theta

(where

\cos\theta \neq 0

) produces:

Pythagorean Identity - Tangent Secant

1 + \tan^2\theta = \sec^2\theta

Learn more about this formula: Pythagorean Identity - Tangent Secant →

Browse all trigonometry formulas →

Dividing by

\sin^2\theta

(where

\sin\theta \neq 0

) produces:

Pythagorean Identity - Cotangent Cosecant

1 + \cot^2\theta = \csc^2\theta

Learn more about this formula: Pythagorean Identity - Cotangent Cosecant →

Browse all trigonometry formulas →

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°

means a full counterclockwise rotation (

360°

) followed by an additional

40°

— the terminal side lands at the same position as

40°

. An angle of

-150°

means a clockwise rotation of

150°

. An angle of

1000°

means two full rotations (

720°

) plus

280°

, so the terminal side coincides with that of

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

\theta

, the angle

\theta + 360°

, the angle

\theta - 360°

, and the angle

\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(\theta + 2\pi) = \sin\theta

is not an algebraic identity to be proved — it is a geometric inevitability. After a full rotation of

2\pi

radians, the point on the circle returns to its starting position, and the

y

-coordinate (sine) is unchanged. The same holds for cosine and for all functions derived from them. Tangent and cotangent repeat after just

\pi

radians because the ratio

\frac{y}{x}

returns to its original value after a half rotation (both

x

and

y

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

\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

P

moving counterclockwise around the unit circle at a constant rate. At any moment, the angle

\theta

swept from the positive

x

-axis determines the position

P = (\cos\theta, \sin\theta)

. Now track just the

y

-coordinate as

\theta

increases. It starts at

0

(when

\theta = 0

), rises to

1

(at

\theta = \frac{\pi}{2}

), returns to

0

(at

\theta = \pi

), drops to

-1

(at

\theta = \frac{3\pi}{2}

), and returns to

0

(at

\theta = 2\pi

). Then the pattern repeats.

Plot this

y

-coordinate against

\theta

, and the result is the familiar wave shape of the sine graph. The

x

-coordinate, tracked similarly, produces the cosine graph — the same wave, shifted by

\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

and

1

: coordinates on a unit circle must lie in

[-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 (

2\pi

radians).

Summary: The Sixteen Standard Angles

The unit circle's computational payoff is a complete reference table for the sixteen standard angles — every multiple of

30°

and

45°

from

0°

330°

. Each row lists the angle in both degree and radian form, the exact

(\cos\theta, \sin\theta)

coordinate pair, and the quadrant. The coordinates are built from only five magnitudes —

0

\frac{1}{2}

\frac{\sqrt{2}}{2}

\frac{\sqrt{3}}{2}

1

— arranged with the correct signs by quadrant. This table underlies every standard-angle evaluation throughout trigonometry.

Angle (degrees)	Angle (radians)	(cos θ, sin θ)	Quadrant
0°	0	(1, 0)	axis (+x)
30°	π/6	(√3/2, 1/2)	I
45°	π/4	(√2/2, √2/2)	I
60°	π/3	(1/2, √3/2)	I
90°	π/2	(0, 1)	axis (+y)
120°	2π/3	(−1/2, √3/2)	II
135°	3π/4	(−√2/2, √2/2)	II
150°	5π/6	(−√3/2, 1/2)	II
180°	π	(−1, 0)	axis (−x)
210°	7π/6	(−√3/2, −1/2)	III
225°	5π/4	(−√2/2, −√2/2)	III
240°	4π/3	(−1/2, −√3/2)	III
270°	3π/2	(0, −1)	axis (−y)
300°	5π/3	(1/2, −√3/2)	IV
315°	7π/4	(√2/2, −√2/2)	IV
330°	11π/6	(√3/2, −1/2)	IV

Unit Circle FAQ

Why is the unit circle radius exactly 1?

Because a radius of 1 makes the coordinates of a point equal the trigonometric values directly. In general the coordinates are r cos θ and r sin θ; setting r to 1 removes the scaling factor entirely, so the x-coordinate is cos θ and the y-coordinate is sin θ. Any other radius would require dividing by r every time.Read more →

Which coordinate is sine and which is cosine?

Cosine is the x-coordinate and sine is the y-coordinate, in that order: the point is written (cos θ, sin θ). Sine feels like it should come first because of the alphabet and the SOH CAH TOA order, but it is second. Reversing the pair reflects every point across the diagonal and gives wrong values everywhere.Read more →

Which trigonometric functions are positive in each quadrant?

All three are positive in the first quadrant. Only sine stays positive in the second, only tangent in the third, and only cosine in the fourth. The signs follow from the coordinates: x is negative on the left half of the circle and y is negative on the bottom half, and tangent is their quotient.Read more →

What are coterminal angles?

Two angles are coterminal when they finish at the same terminal side, differing by a whole number of full rotations. So 45 degrees and 405 degrees are coterminal. As rotations they are different motions, but as positions they coincide, which means they produce identical values for every trigonometric function.Read more →

Do you have to memorise the whole unit circle?

No. Three first-quadrant angles carry it: 30, 45 and 60 degrees. Everything else follows by reflecting those values into the other quadrants and attaching the correct sign for that quadrant. Reference angles do the reflecting, so learning three exact values plus the sign pattern replaces memorising sixteen points.Read more →