How are sine and cosine defined on the unit circle?

On the unit circle, the terminal point of angle theta has coordinates (cos theta, sin theta). The x-coordinate is cos theta and the y-coordinate is sin theta. This definition works for any real angle, positive or negative, including angles beyond 360 degrees, and extends trigonometry beyond the right-triangle setting.

What is the difference between reciprocal and quotient identities?

Reciprocal identities express one function as 1 divided by another, like sec theta equals 1 over cos theta. Quotient identities express one function as a ratio of two others, like tan theta equals sin theta over cos theta. Reciprocal identities pair functions in three couples, while quotient identities show how the four remaining functions all build from sin and cos.

Why does tangent have a period of 180 degrees but sine has 360?

Tangent is the ratio sin theta over cos theta. When theta increases by 180 degrees, both sin and cos flip sign, and a negative divided by a negative is positive. The two sign flips cancel inside the ratio, so tan returns to its original value after only half a turn. Sine and cosine themselves need a full 360-degree turn to return.

How do you find a trig function value for any angle?

First, reduce the angle to its reference angle, which is the acute angle between the terminal ray and the x-axis. Compute or recall the function value for that acute angle. Finally, determine the sign from which quadrant the original angle terminates in. The standard ASTC pattern fixes the sign for each function in each quadrant.

Basic Trig Identities Explorer | Unit Circle Visualizer

Getting Started

The explorer opens centered on the sine function. Three controls drive everything you see:

• Drag the blue dot on the circle to rotate θ
• Use the slider below the circle for precise multi-turn rotation
• Toggle deg / rad for angle units

The graph on the right plots the active function against θ in real time. As you move the dot, the terminal point P traces the unit circle and the dot on the graph tracks the function value.

Switching Between the Six Functions

Two controls swap the active function:

• The tab strip above the circle shows sin, cos, tan, csc, sec, cot in order
• The formula table below the circle is also clickable — tap any row to make it active

When you switch, the diagram redraws to show the correct leg or legs and the graph swaps to that function's curve. The URL updates with a ?fn= query parameter, so any function view can be shared as a direct link.

Stepping Through the Derivation

Each function has a five-stage derivation accessed through the Prev / Play / Next controls:

1. Place the angle — the ray rotates and P appears on the unit circle
2. Identify the leg or legs — vertical for sine, horizontal for cosine, both for tangent
3. Read the value — leg length for sin and cos, a ratio for tan and cot, a reciprocal for csc and sec
4. Sign and range — the quadrant logic and the reference angle
5. Periodicity — drag past 360° to see the spiral arc and coterminal ghost dots

Use Play to auto-advance or step manually with Prev and Next. The rule and description update at each stage.

Reading the Unit Circle Display

The unit circle on the left shows several elements that build up across the steps:

• The red ray from the origin marks the current angle

\theta

• The blue point P sits at

(\cos\theta, \sin\theta)

• The signed leg — vertical, horizontal, or both — shows the function's geometric meaning
• A small arc near the origin marks the reference angle once revealed

Watch how each leg flips sign as P crosses an axis — the vertical leg is sine, the horizontal one cosine. That sign flip is the geometric origin of all four quadrant sign rules.

Reading the Graph

The graph on the right plots the active function across multiple periods. Key features to watch:

• A blue dot tracks the function value at the current θ
• Vertical dashed lines mark asymptotes — at 90° and 270° for tan and sec, at 0°, 180°, and 360° for cot and csc
• Ghost dots highlight coterminal angles where the function takes the same value

The curve's range is fixed per function: bounded between

-1

and

1

for

\sin\theta

and

\cos\theta

, unbounded for the other four.

Exploring Periodicity

The slider sweeps past 360° to reveal what makes trig functions cyclic. As θ exceeds one full turn:

• The ray keeps rotating but P returns to the same circle position
• The arc near the origin spirals outward to count the rotations
• Ghost dots appear on the graph at every coterminal angle

For sin, cos, csc, and sec, this means

f(\theta + 360°) = f(\theta)

. For tan and cot, the period is shorter —

f(\theta + 180°) = f(\theta)

— visible as twice the repetition rate on the graph.

The Live Formula Table

Below the circle, a table lists all six functions with their current values at θ:

• The Function column names the function with

(\theta)

notation
• The Reading column states the geometric or algebraic recipe — vertical leg,

\sin\theta / \cos\theta

,

1 / \sin\theta

, and so on
• The Value at θ column updates as you drag

Watch how reciprocal pairs move together: when

\sin\theta

is small,

\csc\theta

is large. When

\cos\theta = 0

,

\sec\theta

blows up. The table makes these relationships numerical and concrete.

What Are the Basic Trigonometric Identities?

The basic identities relate the six trig functions to each other through simple ratios. They fall into two groups.

Reciprocal identities:

\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}

Quotient identities:

\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Together they mean only two of the six functions — usually

\sin\theta

and

\cos\theta

— are truly independent. The other four are algebraic combinations of those two.

For broader coverage, see trigonometric identities theory.

Defining Trig Functions on the Unit Circle

On the unit circle, the terminal point of angle θ sits at coordinates

(\cos\theta, \sin\theta)

. From this single fact, every trig function follows:

•

\sin\theta

is the y-coordinate of P
•

\cos\theta

is the x-coordinate of P
•

\tan\theta = \sin\theta / \cos\theta

is the slope of the ray
•

\csc\theta

,

\sec\theta

,

\cot\theta

are reciprocals of the above

This definition extends trig beyond right triangles to any real angle — positive or negative, more than 360°, in any quadrant. For the right-triangle perspective, see right triangle trigonometry.

Periodicity and Reference Angles

Two facts make trig functions usable for any angle, no matter how large.

Periodicity means values repeat at a fixed interval. Sin, cos, csc, and sec have period

360°

(or

2\pi

). Tan and cot have period

180°

(or

\pi

) because dividing two functions that both flip sign produces a function that does not.

Reference angle is the acute angle between the terminal ray and the x-axis. It reduces any angle to a Q1 calculation — once you know

|\sin 30°| = 0.5

, you know

|\sin\theta| = 0.5

for every coterminal or reflected angle. The quadrant supplies the sign.

For deeper coverage, see periodicity and reference angles.

Related Concepts and Tools

Explore connected topics:

Trigonometric functions graphs — adjustable amplitude, period, and phase shift for each function

Functions signs by quadrant — focused view of the four quadrant sign patterns

Angle types explorer — acute, obtuse, reflex, and coterminal classification

Pythagorean identity — the

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

relationship from the unit circle

Double angle identities — geometric derivations for

\sin 2\theta

and

\cos 2\theta

Unit circle — the geometric setup every identity here is read from

Angle explorer — quadrants, reference angles and coterminal angles for any input angle

Half angle identities — geometric derivations for

\sin(\theta/2)

and

\cos(\theta/2)

Negative angle identities — parity of the six functions under

\theta \to -\theta

Supplementary angle identities — behaviour of the six functions under

\theta \to \pi - \theta

Sine on the Unit Circle

Sine is the y-coordinate of the terminal point P: as the red ray rotates through

\theta

,

\sin\theta

is the signed height of P above the x-axis, bounded between

-1

and

1

.

Sine paired with its curve, frozen at 50°

The vertical leg of the unit-circle triangle IS the sine; the curve on the right tracks it as the ray turns.

The explorer derives it in five stages: place the angle, drop the vertical leg, trace on the graph, reference angle, and periodicity.

Sine is positive in Quadrants I–II (P above the axis), negative in III–IV, odd (

\sin(-\theta) = -\sin\theta

), and has period

360°

. Its reciprocal is cosecant, and it pairs with cosine to build every other function.

Cosine on the Unit Circle

Cosine is the x-coordinate of the terminal point P: the signed horizontal distance from the origin to the foot of P, bounded between

-1

and

1

.

Cosine paired with its curve, frozen at 50°

The horizontal leg IS the cosine; the curve on the right follows the x-projection of the moving point.

The five-stage derivation: place the angle, project onto the x-axis, trace on the graph, reference angle, and periodicity.

Cosine is positive in Quadrants I and IV (P to the right of the y-axis), even (

\cos(-\theta) = \cos\theta

), and has period

360°

. Its reciprocal is secant; together with sine it generates the four remaining functions.

Tangent on the Unit Circle

Tangent is the ratio of the two legs:

\tan\theta = \sin\theta / \cos\theta

— the slope of the red ray.

Tangent paired with its curve, frozen at 50°

Tangent is the ratio of the legs — vertical over horizontal — with asymptotes wherever the horizontal leg vanishes.

The derivation: place the angle, read both legs, form the ratio, sign by quadrant, and periodicity.

Tangent is unbounded, diverging at

90°

and

270°

where the cosine leg vanishes — the dashed asymptotes on the graph. Its reciprocal is cotangent, and its period is only

180°

.

Cosecant on the Unit Circle

Cosecant is the reciprocal of sine:

\csc\theta = 1 / \sin\theta

, read from the same vertical leg as sine and then inverted.

Cosecant paired with its curve, frozen at 50°

Cosecant flips the vertical leg into its reciprocal: U-branches blowing up wherever sine shrinks to zero.

The derivation: place the angle, identify the vertical leg, take the reciprocal, range and sign, and periodicity.

Wherever it is defined,

|\csc\theta| \ge 1

— a small sine makes a large cosecant. It diverges at

0°

,

180°

, and

360°

, always matches sine's sign, and shares sine's

360°

period.

Secant on the Unit Circle

Secant is the reciprocal of cosine:

\sec\theta = 1 / \cos\theta

, built from the horizontal leg of cosine.

Secant paired with its curve, frozen at 50°

Secant flips the horizontal leg: the same U-branch architecture, with poles at cosine’s zeros.

The derivation: place the angle, identify the horizontal leg, take the reciprocal, range and sign, and periodicity.

Like its partner cosecant, secant satisfies

|\sec\theta| \ge 1

wherever defined. It diverges at

90°

and

270°

, matches cosine's sign, and repeats every

360°

.

Cotangent on the Unit Circle

Cotangent is the inverted ratio:

\cot\theta = \cos\theta / \sin\theta = 1 / \tan\theta

— the run over the rise of the red ray.

Cotangent paired with its curve, frozen at 50°

Cotangent is horizontal over vertical — tangent inverted, falling where tangent climbs.

The derivation: place the angle, read both legs, form the ratio, sign by quadrant, and periodicity.

Cotangent diverges where the sine leg vanishes —

0°

,

180°

,

360°

— exactly where tangent is zero, and shares tangent's

180°

period and quadrant sign pattern.

Sine, Step 1: Place the Angle

The derivation of sine starts by rotating the red ray counterclockwise from the positive x-axis through

\theta

. The terminal point P appears where the ray meets the unit circle.

Sine — Step 1: the angle placed

The ray parks at 50°: circle on the left, the sine graph on the right, nothing highlighted yet.

Everything that follows reads geometry off P — placing the angle in standard position is what turns "an angle" into "a point whose coordinates we can measure."

Sine, Step 2: Drop the Vertical Leg

From P, a perpendicular drops to the x-axis. The signed length of that vertical leg is

\sin\theta

: positive when P is above the axis, negative below.

Sine — Step 2: the leg highlighted

The vertical leg lights up on the circle — the geometric quantity that IS sin θ.

This is the geometric definition — sine is not a formula but a length with a sign, which is why it can never exceed the circle's radius of

1

.

Sine, Step 3: Trace on the Graph

Plot

\theta

horizontally and

\sin\theta

vertically: the tracking dot on the curve sits at exactly the same height as P's y-coordinate on the circle.

Sine — Step 3: pinned to the graph

The tracking dot pins the graph at θ = 50°, tying the circle reading to the curve.

Dragging the dot shows the circle and the wave are the same information in two pictures — the sine wave is the circle's height unrolled along the angle axis.

Sine, Step 4: Reference Angle

The green arc marks the reference angle — the acute angle between the ray and the nearest x-axis. Its sine matches

\sin\theta

in magnitude; the quadrant fixes the sign.

Sine — Step 4: reference angle at 140°

At 140° the reference-angle arc shows the same magnitude — the quadrant alone decides the sign.

At the frozen

140°

the reference angle is

40°

:

\sin 140° = \sin 40°

, positive because Quadrant II keeps the vertical leg above the axis.

Sine, Step 5: Periodicity

Past

360°

the ray keeps rotating — the red arc spirals outward counting turns — but P returns to the same position, so

\sin(\theta + 360°) = \sin\theta

.

Sine — Step 5: one turn later

At 410° the spiral overlays the 50° ghost: one full turn later, the value returns exactly.

The ghost dots on the graph mark coterminal angles at the same height: the frozen

410°

repeats the value of

50°

exactly one turn earlier.

Cosine, Step 1: Place the Angle

The derivation of cosine begins the same way every function does: rotate the red ray through

\theta

and mark the terminal point P on the unit circle.

Cosine — Step 1: the angle placed

The ray parks at 50°: circle on the left, the cosine graph on the right, nothing highlighted yet.

With P fixed, cosine will come from the horizontal direction — the complementary reading to sine's vertical one.

Cosine, Step 2: Project onto the X-Axis

Project P straight down (or up) onto the x-axis. The signed distance from the origin to that foot is

\cos\theta

: positive to the right of the origin, negative to the left.

Cosine — Step 2: the leg highlighted

The horizontal projection lights up — the geometric quantity that IS cos θ.

The amber horizontal leg is the whole definition — cosine measures how far around the circle the angle has carried P in the x-direction.

Cosine, Step 3: Trace on the Graph

Plot

\theta

horizontally and

\cos\theta

vertically. The tracking dot follows the length of the amber leg as the ray sweeps.

Cosine — Step 3: pinned to the graph

The tracking dot pins the graph at θ = 50°, tying the circle reading to the curve.

The result is the cosine wave — the same shape as sine but starting at its maximum of

1

, because at

\theta = 0°

the point P sits fully to the right.

Cosine, Step 4: Reference Angle

The reference angle — the acute angle between the ray and the nearest x-axis — carries cosine's magnitude; the quadrant decides the sign.

Cosine — Step 4: reference angle at 140°

At 140° the reference-angle arc shows the same magnitude — the quadrant alone decides the sign.

At the frozen

140°

: reference angle

40°

, so

\cos 140° = -\cos 40°

— negative, because Quadrant II puts the foot of P left of the origin.

Cosine, Step 5: Periodicity

One full turn brings P back exactly:

\cos(\theta + 360°) = \cos\theta

. The spiral arc counts rotations while the value cycles.

Cosine — Step 5: one turn later

At 410° the spiral overlays the 50° ghost: one full turn later, the value returns exactly.

Ghost dots on the curve mark the coterminal angles — at

410°

the amber leg is identical to the one at

50°

.

Tangent, Step 1: Place the Angle

The derivation of tangent starts from the same standard position: red ray through

\theta

, terminal point P on the circle.

Tangent — Step 1: the angle placed

The ray parks at 50°: circle on the left, the tangent graph on the right, nothing highlighted yet.

Tangent will need both coordinates of P, so this single placement feeds two measurements at once.

Tangent, Step 2: Read Both Legs

Both legs light up: the blue vertical leg is

\sin\theta

and the amber horizontal leg is

\cos\theta

. Tangent uses the pair.

Tangent — Step 2: the leg highlighted

Both legs light up: tangent will be their ratio, vertical over horizontal.

Seeing the two legs together is the point of this step — tangent is not a new measurement but a relationship between the two existing ones.

Tangent, Step 3: Form the Ratio

Divide:

\tan\theta = \sin\theta / \cos\theta

— rise over run, the slope of the red ray. The graph plots that ratio.

Tangent — Step 3: pinned to the graph

The tracking dot pins the graph at θ = 50°, tying the circle reading to the curve.

Where the amber leg shrinks to zero —

90°

and

270°

— the division blows up: the dashed vertical asymptotes on the graph.

Tangent, Step 4: Sign by Quadrant

Tangent is positive where the legs agree in sign — Quadrants I and III — and negative where they differ — Quadrants II and IV.

Tangent — Step 4: reference angle at 140°

At 140° the reference-angle arc shows the same magnitude — the quadrant alone decides the sign.

At the frozen

140°

the blue leg is positive and the amber leg negative, so

\tan 140°

is negative: the tracking dot sits below the axis.

Tangent, Step 5: Periodicity

Tangent repeats every

180°

— twice as fast as sine and cosine — because a half turn flips both legs and the two sign changes cancel in the ratio:

\tan(\theta + 180°) = \tan\theta

.

Tangent — Step 5: one turn later

At 410° the spiral overlays the 50° ghost: one full turn later, the value returns exactly.

The graph fits twice as many periods into the same span; the ghost dots mark repeats every half turn, not just every full one.

Cosecant, Step 1: Place the Angle

The derivation of cosecant begins with the standard placement: red ray through

\theta

, terminal point P on the unit circle.

Cosecant — Step 1: the angle placed

The ray parks at 50°: circle on the left, the cosecant graph on the right, nothing highlighted yet.

Cosecant is built on sine, so the vertical direction is where this derivation is headed.

Cosecant, Step 2: Identify the Vertical Leg

The blue vertical leg from P to the x-axis is

\sin\theta

— the same leg the sine derivation uses.

Cosecant — Step 2: the leg highlighted

The vertical leg lights up — cosecant is about to flip it into a reciprocal.

Cosecant adds nothing geometric at this stage; it borrows sine's measurement and prepares to invert it.

Cosecant, Step 3: Take the Reciprocal

Invert the leg:

\csc\theta = 1 / \sin\theta

. The graph shows the consequence — where sine is small, cosecant is huge.

Cosecant — Step 3: pinned to the graph

The tracking dot pins the graph at θ = 50°, tying the circle reading to the curve.

At

0°

,

180°

, and

360°

the leg vanishes entirely and cosecant diverges: the dashed asymptotes sit exactly at sine's zeros.

Cosecant, Step 4: Range and Sign

Because

|\sin\theta| \le 1

, its reciprocal satisfies

|\csc\theta| \ge 1

wherever defined — the curve never enters the band between

-1

and

1

.

Cosecant — Step 4: reference angle at 140°

At 140° the reference-angle arc shows the same magnitude — the quadrant alone decides the sign.

The sign always matches sine's: positive in Quadrants I–II, negative in III–IV. The frozen

140°

gives a positive cosecant just above

1.5

.

Cosecant, Step 5: Periodicity

Cosecant inherits its cycle from sine:

\csc(\theta + 360°) = \csc\theta

. The spiral counts turns while the value repeats.

Cosecant — Step 5: one turn later

At 410° the spiral overlays the 50° ghost: one full turn later, the value returns exactly.

Ghost dots mark the coterminal repeats — every branch of the curve returns identically after each full rotation.

Secant, Step 1: Place the Angle

The derivation of secant starts identically: red ray through

\theta

, terminal point P on the circle.

Secant — Step 1: the angle placed

The ray parks at 50°: circle on the left, the secant graph on the right, nothing highlighted yet.

Secant builds on cosine, so the horizontal leg is the one to watch.

Secant, Step 2: Identify the Horizontal Leg

The amber horizontal leg from the origin to the foot of P is

\cos\theta

— the cosine measurement, reused.

Secant — Step 2: the leg highlighted

The horizontal leg lights up — secant is its reciprocal.

As with cosecant, no new geometry appears: secant is an algebraic move performed on an existing leg.

Secant, Step 3: Take the Reciprocal

Invert the leg:

\sec\theta = 1 / \cos\theta

. The graph diverges where the amber leg vanishes — at

90°

and

270°

, cosine's zeros.

Secant — Step 3: pinned to the graph

The tracking dot pins the graph at θ = 50°, tying the circle reading to the curve.

Those are the same asymptote positions as tangent's, since both divide by

\cos\theta

.

Secant, Step 4: Range and Sign

Since

|\cos\theta| \le 1

, the reciprocal obeys

|\sec\theta| \ge 1

— the curve stays outside the unit band, touching it only at

\theta = 0°

,

180°

,

360°

.

Secant — Step 4: reference angle at 140°

At 140° the reference-angle arc shows the same magnitude — the quadrant alone decides the sign.

The sign follows cosine: positive in Quadrants I and IV, negative in II and III. At the frozen

140°

, secant is negative.

Secant, Step 5: Periodicity

Secant repeats with cosine's full-turn cycle:

\sec(\theta + 360°) = \sec\theta

, spiral growing while the value loops.

Secant — Step 5: one turn later

At 410° the spiral overlays the 50° ghost: one full turn later, the value returns exactly.

Ghost dots on the graph mark each coterminal angle where the branch pattern recurs.

Cotangent, Step 1: Place the Angle

The derivation of cotangent opens with the shared first move: red ray through

\theta

, terminal point P.

Cotangent — Step 1: the angle placed

The ray parks at 50°: circle on the left, the cotangent graph on the right, nothing highlighted yet.

Like tangent, cotangent will read both legs of P — just in the opposite order.

Cotangent, Step 2: Read Both Legs

Both legs appear: the blue vertical

\sin\theta

and the amber horizontal

\cos\theta

.

Cotangent — Step 2: the leg highlighted

Both legs light up: cotangent takes horizontal over vertical.

The pair is the same as tangent's; the difference is entirely in which leg goes on top of the ratio.

Cotangent, Step 3: Form the Ratio

Divide the other way:

\cot\theta = \cos\theta / \sin\theta = 1 / \tan\theta

— run over rise. The graph diverges where the sine leg vanishes:

0°

,

180°

,

360°

.

Cotangent — Step 3: pinned to the graph

The tracking dot pins the graph at θ = 50°, tying the circle reading to the curve.

Cotangent's asymptotes sit exactly where tangent crosses zero, and vice versa — the two curves interlock.

Cotangent, Step 4: Sign by Quadrant

Cotangent shares tangent's sign pattern: positive in Quadrants I and III where the legs agree, negative in II and IV where they differ.

Cotangent — Step 4: reference angle at 140°

At 140° the reference-angle arc shows the same magnitude — the quadrant alone decides the sign.

At the frozen

140°

the legs disagree in sign, so

\cot 140°

is negative — the dot sits below the axis.

Cotangent, Step 5: Periodicity

Like tangent, cotangent repeats every half turn:

\cot(\theta + 180°) = \cot\theta

, because both legs flip sign together and the flips cancel in the ratio.

Cotangent — Step 5: one turn later

At 410° the spiral overlays the 50° ghost: one full turn later, the value returns exactly.

The graph packs a full copy of the curve into each

180°

span, with ghost dots marking every repeat.