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Basic Trigonometric Identities


How to use. Drag the blue dot on the circle (or use the slider) to change θ. The slider sweeps through multiple turns — past 360° the arc spirals outward and the graph shows ghost dots at coterminal angles. Toggle deg/rad for units, use the tabs below to pick a function, and the Prev / Play / Next controls to step through.
sin(θ)
θ30°
Unit circle
Unit circle scenexyOθsin θref = 30°P
Function graph
Function graphturn-360°0°360°720°1.10−1.1y = sin θ-330°390°0.50030°
Step 1 of 5

sin θ

0.500

Derivation
1Place the angle

Rotate the ray from the positive x-axis through θ counterclockwise. The terminal point P sits on the unit circle. Full treatment · All about sine

Function
Reading
Value at θ

Getting Started
Switching Between the Six Functions
Stepping Through the Derivation
Reading the Unit Circle Display
Reading the Graph
Exploring Periodicity
The Live Formula Table
What Are the Basic Trigonometric Identities?
Defining Trig Functions on the Unit Circle
Periodicity and Reference Angles
Related Concepts and Tools
Sine on the Unit Circle
Cosine on the Unit Circle
Tangent on the Unit Circle
Cosecant on the Unit Circle
Secant on the Unit Circle
Cotangent on the Unit Circle
Sine, Step 1: Place the Angle
Sine, Step 2: Drop the Vertical Leg
Sine, Step 3: Trace on the Graph
Sine, Step 4: Reference Angle
Sine, Step 5: Periodicity
Cosine, Step 1: Place the Angle
Cosine, Step 2: Project onto the X-Axis
Cosine, Step 3: Trace on the Graph
Cosine, Step 4: Reference Angle
Cosine, Step 5: Periodicity
Tangent, Step 1: Place the Angle
Tangent, Step 2: Read Both Legs
Tangent, Step 3: Form the Ratio
Tangent, Step 4: Sign by Quadrant
Tangent, Step 5: Periodicity
Cosecant, Step 1: Place the Angle
Cosecant, Step 2: Identify the Vertical Leg
Cosecant, Step 3: Take the Reciprocal
Cosecant, Step 4: Range and Sign
Cosecant, Step 5: Periodicity
Secant, Step 1: Place the Angle
Secant, Step 2: Identify the Horizontal Leg
Secant, Step 3: Take the Reciprocal
Secant, Step 4: Range and Sign
Secant, Step 5: Periodicity
Cotangent, Step 1: Place the Angle
Cotangent, Step 2: Read Both Legs
Cotangent, Step 3: Form the Ratio
Cotangent, Step 4: Sign by Quadrant
Cotangent, Step 5: Periodicity






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 θ\theta
• The blue point P sits at (cos⁡θ,sin⁡θ)(\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 −1-1 and 11 for sin⁡θ\sin\theta and cos⁡θ\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(θ+360°)=f(θ)f(\theta + 360°) = f(\theta). For tan and cot, the period is shorter — f(θ+180°)=f(θ)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 (θ)(\theta) notation
• The Reading column states the geometric or algebraic recipe — vertical leg, sin⁡θ/cos⁡θ\sin\theta / \cos\theta, 1/sin⁡θ1 / \sin\theta, and so on
• The Value at θ column updates as you drag

Watch how reciprocal pairs move together: when sin⁡θ\sin\theta is small, csc⁡θ\csc\theta is large. When cos⁡θ=0\cos\theta = 0, sec⁡θ\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⁡θ=1sin⁡θsec⁡θ=1cos⁡θcot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}


Quotient identities:

tan⁡θ=sin⁡θcos⁡θcot⁡θ=cos⁡θsin⁡θ\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⁡θ\sin\theta and cos⁡θ\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⁡θ,sin⁡θ)(\cos\theta, \sin\theta). From this single fact, every trig function follows:

• sin⁡θ\sin\theta is the y-coordinate of P
• cos⁡θ\cos\theta is the x-coordinate of P
• tan⁡θ=sin⁡θ/cos⁡θ\tan\theta = \sin\theta / \cos\theta is the slope of the ray
• csc⁡θ\csc\theta, sec⁡θ\sec\theta, cot⁡θ\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°360° (or 2π2\pi). Tan and cot have period 180°180° (or π\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|\sin 30°| = 0.5, you know ∣sin⁡θ∣=0.5|\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⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 relationship from the unit circle

Double angle identities — geometric derivations for sin⁡2θ\sin 2\theta and cos⁡2θ\cos 2\theta

Sine on the Unit Circle

Sine is the y-coordinate of the terminal point P: as the red ray rotates through θ\theta, sin⁡θ\sin\theta is the signed height of P above the x-axis, bounded between −1-1 and 11.
50°y = sin θ
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⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta), and has period 360°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 −1-1 and 11.
50°y = cos θ
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⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta), and has period 360°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⁡θ=sin⁡θ/cos⁡θ\tan\theta = \sin\theta / \cos\theta — the slope of the red ray.
50°y = tan θ
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°90° and 270°270° where the cosine leg vanishes — the dashed asymptotes on the graph. Its reciprocal is cotangent, and its period is only 180°180°.

Cosecant on the Unit Circle

Cosecant is the reciprocal of sine: csc⁡θ=1/sin⁡θ\csc\theta = 1 / \sin\theta, read from the same vertical leg as sine and then inverted.
50°y = csc θ
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⁡θ∣≥1|\csc\theta| \ge 1 — a small sine makes a large cosecant. It diverges at 0°0°, 180°180°, and 360°360°, always matches sine's sign, and shares sine's 360°360° period.

Secant on the Unit Circle

Secant is the reciprocal of cosine: sec⁡θ=1/cos⁡θ\sec\theta = 1 / \cos\theta, built from the horizontal leg of cosine.
50°y = sec θ
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⁡θ∣≥1|\sec\theta| \ge 1 wherever defined. It diverges at 90°90° and 270°270°, matches cosine's sign, and repeats every 360°360°.

Cotangent on the Unit Circle

Cotangent is the inverted ratio: cot⁡θ=cos⁡θ/sin⁡θ=1/tan⁡θ\cot\theta = \cos\theta / \sin\theta = 1 / \tan\theta — the run over the rise of the red ray.
50°y = cot θ
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°0°, 180°180°, 360°360° — exactly where tangent is zero, and shares tangent's 180°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 θ\theta. The terminal point P appears where the ray meets the unit circle.
50°y = sin θ
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⁡θ\sin\theta: positive when P is above the axis, negative below.
50°y = sin θ
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 11.

Sine, Step 3: Trace on the Graph

Plot θ\theta horizontally and sin⁡θ\sin\theta vertically: the tracking dot on the curve sits at exactly the same height as P's y-coordinate on the circle.
50°y = sin θ
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⁡θ\sin\theta in magnitude; the quadrant fixes the sign.
140°y = sin θ
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°140° the reference angle is 40°40°: sin⁡140°=sin⁡40°\sin 140° = \sin 40°, positive because Quadrant II keeps the vertical leg above the axis.

Sine, Step 5: Periodicity

Past 360°360° the ray keeps rotating — the red arc spirals outward counting turns — but P returns to the same position, so sin⁡(θ+360°)=sin⁡θ\sin(\theta + 360°) = \sin\theta.
410°y = sin θ
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°410° repeats the value of 50°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 θ\theta and mark the terminal point P on the unit circle.
50°y = cos θ
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⁡θ\cos\theta: positive to the right of the origin, negative to the left.
50°y = cos θ
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 θ\theta horizontally and cos⁡θ\cos\theta vertically. The tracking dot follows the length of the amber leg as the ray sweeps.
50°y = cos θ
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 11, because at θ=0°\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.
140°y = cos θ
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°140°: reference angle 40°40°, so cos⁡140°=−cos⁡40°\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⁡(θ+360°)=cos⁡θ\cos(\theta + 360°) = \cos\theta. The spiral arc counts rotations while the value cycles.
410°y = cos θ
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°410° the amber leg is identical to the one at 50°50°.

Tangent, Step 1: Place the Angle

The derivation of tangent starts from the same standard position: red ray through θ\theta, terminal point P on the circle.
50°y = tan θ
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⁡θ\sin\theta and the amber horizontal leg is cos⁡θ\cos\theta. Tangent uses the pair.
50°y = tan θ
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⁡θ=sin⁡θ/cos⁡θ\tan\theta = \sin\theta / \cos\theta — rise over run, the slope of the red ray. The graph plots that ratio.
50°y = tan θ
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°90° and 270°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.
140°y = tan θ
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°140° the blue leg is positive and the amber leg negative, so tan⁡140°\tan 140° is negative: the tracking dot sits below the axis.

Tangent, Step 5: Periodicity

Tangent repeats every 180°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⁡(θ+180°)=tan⁡θ\tan(\theta + 180°) = \tan\theta.
410°y = tan θ
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 θ\theta, terminal point P on the unit circle.
50°y = csc θ
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⁡θ\sin\theta — the same leg the sine derivation uses.
50°y = csc θ
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⁡θ=1/sin⁡θ\csc\theta = 1 / \sin\theta. The graph shows the consequence — where sine is small, cosecant is huge.
50°y = csc θ
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°0°, 180°180°, and 360°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⁡θ∣≤1|\sin\theta| \le 1, its reciprocal satisfies ∣csc⁡θ∣≥1|\csc\theta| \ge 1 wherever defined — the curve never enters the band between −1-1 and 11.
140°y = csc θ
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°140° gives a positive cosecant just above 1.51.5.

Cosecant, Step 5: Periodicity

Cosecant inherits its cycle from sine: csc⁡(θ+360°)=csc⁡θ\csc(\theta + 360°) = \csc\theta. The spiral counts turns while the value repeats.
410°y = csc θ
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 θ\theta, terminal point P on the circle.
50°y = sec θ
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⁡θ\cos\theta — the cosine measurement, reused.
50°y = sec θ
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⁡θ=1/cos⁡θ\sec\theta = 1 / \cos\theta. The graph diverges where the amber leg vanishes — at 90°90° and 270°270°, cosine's zeros.
50°y = sec θ
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⁡θ\cos\theta.

Secant, Step 4: Range and Sign

Since ∣cos⁡θ∣≤1|\cos\theta| \le 1, the reciprocal obeys ∣sec⁡θ∣≥1|\sec\theta| \ge 1 — the curve stays outside the unit band, touching it only at θ=0°\theta = 0°, 180°180°, 360°360°.
140°y = sec θ
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°140°, secant is negative.

Secant, Step 5: Periodicity

Secant repeats with cosine's full-turn cycle: sec⁡(θ+360°)=sec⁡θ\sec(\theta + 360°) = \sec\theta, spiral growing while the value loops.
410°y = sec θ
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 θ\theta, terminal point P.
50°y = cot θ
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⁡θ\sin\theta and the amber horizontal cos⁡θ\cos\theta.
50°y = cot θ
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⁡θ=cos⁡θ/sin⁡θ=1/tan⁡θ\cot\theta = \cos\theta / \sin\theta = 1 / \tan\theta — run over rise. The graph diverges where the sine leg vanishes: 0°0°, 180°180°, 360°360°.
50°y = cot θ
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.
140°y = cot θ
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°140° the legs disagree in sign, so cot⁡140°\cot 140° is negative — the dot sits below the axis.

Cotangent, Step 5: Periodicity

Like tangent, cotangent repeats every half turn: cot⁡(θ+180°)=cot⁡θ\cot(\theta + 180°) = \cot\theta, because both legs flip sign together and the flips cancel in the ratio.
410°y = cot θ
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°180° span, with ghost dots marking every repeat.