What angle topics does the explorer cover?

Nine concepts in three groups. Classification covers angle types like acute, right, obtuse, straight, and reflex. Relationships covers complementary, supplementary, vertical, and adjacent angles. Trigonometry covers standard position, reference angles, coterminal angles, special angles, and directed angles.

How do I switch between the nine concepts?

On wide screens, a sidebar on the left lists all concepts grouped by category. On narrow screens, the same list becomes a horizontal scrolling tab strip above the scene. Click any item to load its interactive diagram and explanation.

Can I work with negative angles?

Yes. In the Standard Position and Directed Angles concepts, dragging below the x-axis produces clockwise (negative) rotations. The diagram color switches to amber and the side panel computes both directions, demonstrating that sine is odd and cosine is even.

What is a reference angle and how do I find one?

The reference angle is the acute angle between the terminal side and the nearest x-axis, always between 0 and 90 degrees. The formula depends on the quadrant: theta in Q1, 180 minus theta in Q2, theta minus 180 in Q3, and 360 minus theta in Q4. The Reference Angles concept shows these formulas live.

How do special angles relate to the rest of the tool?

Special angles are the 16 unit-circle positions (0, 30, 45, 60, 90 degrees and their reflections) that have exact trig values like 1/2, sqrt(2)/2, sqrt(3)/2. They appear as snap points in the other concepts and as a clickable ring of dots in the Special Angles view.

Angle Types Explorer: Interactive Visualizer

Q	sin	cos	tan
I	+	+	+
II	+	−	−
III	−	−	+
IV	−	+	−

Key Terms

• Vertex — the common endpoint where the two rays of an angle meet.
• Initial side — the ray from which rotation is measured. In standard position it lies along the positive x-axis.
• Terminal side — the ray reached after rotating by the angle.
• Quadrant — one of four regions

I

,

II

,

III

,

IV

where the terminal side may land.
• Reference angle — the acute angle between the terminal side and the nearest x-axis. Always between

0°

and

90°

.
• Coterminal angles — angles sharing the same terminal side, differing by full rotations (

360° n

).
• Special angles — the

16

unit-circle angles with exact

\sin

,

\cos

,

\tan

values.
• Directed angle — an angle carrying a sign indicating rotation direction: positive for counterclockwise, negative for clockwise.

Choosing a Concept

The tool bundles nine related angle topics. A sidebar (or horizontal scroll on narrow screens) lists them grouped into three categories:

• Classification — Angle Types.
• Relationships — Complementary & Supplementary, Vertical Angles, Adjacent Angles.
• Trigonometry — Standard Position, Reference Angles, Coterminal Angles, Special Angles, Directed Angles.

Click any item to load that concept's interactive scene and explanation. The active item is highlighted in blue. The top bar shows both the tool name and the active concept's title.

Switching Dark and Light Modes

A Light / Dark toggle in the top-right of the panel swaps the color theme across all nine concepts.

What changes with the toggle:
• Background — white in light mode, deep slate in dark mode.
• Text colors — adjusted for contrast.
• Panel borders and dividers — match the active theme.
• Diagram fill colors (blue, amber, green, red, purple accents) remain the same in both modes so visualizations stay readable.

The theme persists while you switch between concepts in the same session.

Dragging to Set Angles

Every scene includes one or more draggable handles. Click and drag a handle to rotate the relevant arm of the angle in real time.

How drag works:
• The handle follows your pointer around the vertex.
• The angle value updates continuously in the side panel.
• Some scenes (Vertical, Adjacent) include two independent handles.
• On Standard Position and Directed Angles, drag below the x-axis to produce negative angles.

Snap-to-special behavior makes it easy to land on common values: as you approach

0°

,

30°

,

45°

,

60°

,

90°

, etc., the handle locks onto the exact value.

Using Preset Buttons and Quick Angles

Some scenes provide preset buttons for fast navigation to canonical angles.

Examples:
• Angle Types — seven buttons (Zero, Acute, Right, Obtuse, Straight, Reflex, Full) jump to representative angles.
• Complementary & Supplementary — a two-tab switch toggles the constraint between summing to

90°

and

180°

.
• Coterminal Angles —

+

and

-

buttons step through full-rotation offsets from

n = -3

to

n = +3

.
• Special Angles — clicking any of the

16

ringed points on the unit circle selects it; a Degrees / Radians / Both toggle controls labels.

Presets are the fastest way to see the boundary cases of each concept.

Reading the Scene Diagrams

Each scene uses a consistent color language to keep the relationships visible at a glance.

Color conventions across scenes:
• Blue — the active or first angle.
• Amber — the partner angle (the second in a pair, or the negative direction).
• Green — right angles, positive sign markers,

\cos

measurements.
• Red — obtuse markers, negative sign markers,

\sin

measurements.
• Purple — shared rays or alternative emphasis (e.g., the common arm of adjacent angles).
• Gray — reference axes, dashed guides, inactive elements.

A small square at a vertex marks an exact

90°

angle in place of the usual curved arc.

Reading the Explanation Panels

The right-hand panel of every concept contains four consistent zones:

• Title — name of the active concept.
• Brief description — one or two paragraphs explaining what the concept means and why it matters in trigonometry.
• Live values — color-coded mini-cards or large numerals showing the current angle, partner angle, sum, or sign data.
• Reference block — a formula card (monospace) or a small table summarizing the rule (e.g., quadrant sign chart, reference-angle formulas, even / odd identities).

Drag the scene and the explanation panel updates immediately. Everything is recomputed from the current angle.

Angle Classifications

The Angle Types concept classifies a single rotation by its measure:

• Zero —

\theta = 0°

, both rays overlap.
• Acute —

0° < \theta < 90°

.
• Right —

\theta = 90°

, marked with a square instead of an arc.
• Obtuse —

90° < \theta < 180°

.
• Straight —

\theta = 180°

, a straight line.
• Reflex —

180° < \theta < 360°

.
• Full —

\theta = 360°

, a complete rotation.

Each type uses its own color and is documented in the side panel. For comprehensive coverage with proofs and examples, see the angle types theory page.

Angle Relationships

Three concepts cover how angles relate when they share a vertex or a transversal.

• Complementary & Supplementary — pairs summing to

90°

or

180°

. The side panel surfaces cofunction identities like

\sin\theta = \cos(90° - \theta)

.
• Vertical Angles — two intersecting lines create two pairs of equal opposite angles. Drag to

90°

to see all four become right angles.
• Adjacent Angles — two angles share a vertex and a common arm (drawn in purple, dashed). Two independent drag handles let you set each angle separately. The angle addition identities sit right under the diagram.

For full proofs, see the angle relationships page.

Trigonometry-Specific Angle Concepts

Five concepts cover angle ideas central to trigonometry.

• Standard Position — angle with vertex at origin and initial side on the positive x-axis. The side panel shows the active quadrant and the signs of

\sin

,

\cos

,

\tan

.
• Reference Angles — the acute angle between the terminal side and the nearest x-axis. The panel shows the formula appropriate to the active quadrant (

\theta

,

180° - \theta

,

\theta - 180°

, or

360° - \theta

).
• Coterminal Angles — same terminal side, different rotation count. Step through

n \in \{-3, ..., 3\}

and watch the spiral marker connect the base angle to its coterminal partner.
• Special Angles — the

16

unit-circle positions whose

\sin

,

\cos

,

\tan

values are exact. Click any point or row in the table to load it.
• Directed Angles — positive (counterclockwise) vs. negative (clockwise) angles, with live verification that

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

(odd) and

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

(even).

Why Angle Classification Matters in Trigonometry

Knowing the type of an angle determines almost every downstream calculation:

• Function signs depend on quadrant, which depends on classification (acute, obtuse, reflex).
• Reference angles reduce any trig evaluation to a first-quadrant computation.
• Coterminal equivalence means

\sin\theta

,

\cos\theta

,

\tan\theta

are periodic; this powers Fourier analysis, wave physics, and signal processing.
• Special angles provide the exact values that appear in proofs, identities, and integrals.
• Directed angles distinguish phase and rotation direction in physics, navigation, and complex numbers.

For applications and worked examples, see the trigonometry foundations page.

Related Concepts and Tools

Continue exploring with these connected resources:

• Angle Explorer — single-angle visualizer focused on type, quadrant, and reference angle.
• Trigonometric Functions Graphs — see how

\sin

,

\cos

,

\tan

and their reciprocals evolve as

\theta

varies.
• Functions Signs by Quadrants — the ASTC rule explained interactively.
• Triangle Explorer — angles inside triangles, with built-in law of sines and law of cosines.
• Unit Circle — geometric setup for every special angle in this tool.
• Double Angle Identities — formulas that combine angles into

2\theta

relationships.

Complementary and Supplementary Angles

Two angles are complementary when their measures sum to

90°

and supplementary when they sum to

180°

. The scene draws the pair as a blue sector

\alpha

and an amber sector

\beta

sharing one vertex, with a two-tab toggle switching the constraint.

Complement and supplement, frozen

One angle shown against both partners: the complement closing 90° and the supplement closing 180°.

The complementary constraint fills a right angle, the supplementary one a straight angle. Dragging the blue handle reallocates the total between

\alpha

and

\beta

while their sum stays fixed — the live sum line under the diagram never changes.

The pair relationship powers the cofunction identities shown in the panel's formula card:

\sin\theta = \cos(90° - \theta)

,

\cos\theta = \sin(90° - \theta)

, and

\tan\theta = \cot(90° - \theta)

.

Vertical Angles

When two straight lines intersect, they create two pairs of vertical angles — opposite angles that are always equal. The blue pair and the amber pair in the scene stay matched no matter how you drag either line.

Vertical angles, frozen

Two lines crossing: the opposite angle pairs are mirror images and always equal.

Why they must be equal: each blue angle and its amber neighbor lie on one straight line, so they are supplementary. Two angles supplementary to the same angle are equal — that is the whole proof, visible live as you drag.

Drag to

90°

and all four angles become equal right angles: the two lines are perpendicular, and the four sectors match exactly.

Adjacent Angles

Adjacent angles share a vertex and one common arm — drawn dashed and purple in the scene — and lie on opposite sides of it without overlapping. Two independent handles set

\alpha

(blue) and

\beta

(amber) separately.

Adjacent angles, frozen

Two angles sharing a vertex and one ray, sitting side by side without overlapping.

Unlike complementary or supplementary pairs, adjacent angles have no fixed total: the combined angle is simply

\alpha + \beta

, updated live under the diagram.

Adjacency is the geometric picture behind the angle addition identities:

\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

computes the sine of the combined angle from the two parts you dragged.

Standard Position

An angle is in standard position when its vertex sits at the origin and its initial side lies along the positive x-axis. Positive angles rotate counterclockwise (blue); negative angles rotate clockwise (amber) — drag below the x-axis to produce them.

Standard position, frozen

Vertex at the origin, initial side fixed on the positive x-axis — the reference frame for every angle in trigonometry.

The quadrant containing the terminal side fixes the signs of all six trigonometric functions: Quadrant I makes everything positive, Quadrant II keeps only sine positive, Quadrant III only tangent, and Quadrant IV only cosine.

Standard position is the convention that makes the rest of trigonometry work: reference angles, coterminal angles, and the unit-circle values all assume it.

Reference Angles

The reference angle of

\theta

is the acute angle between its terminal side and the nearest x-axis — always between

0°

and

90°

, always positive. In the scene it is the amber arc hugging the x-axis while the faint blue arc shows the full rotation.

Reference angle, frozen

The acute gap between the terminal ray and the x-axis — the quadrant-free core of the angle.

The formula depends on the quadrant:

\theta

itself in Quadrant I,

180° - \theta

in Quadrant II,

\theta - 180°

in Quadrant III, and

360° - \theta

in Quadrant IV. The panel highlights the active row as you drag.

Reference angles are the reduction machine of trigonometry: any function of any angle equals plus-or-minus the same function of its reference angle, with the sign taken from the quadrant.

Coterminal Angles

Coterminal angles share the same terminal side while differing by whole rotations:

\theta

and

\theta + 360°n

point the same way for every integer

n

. The scene's stepper walks

n

from

-3

to

+3

and draws the extra rotations as a dashed purple spiral.

Coterminal pair, frozen

Two different rotations ending on one terminal ray — a full turn apart.

Because the terminal side is identical, every trigonometric value is identical too:

\sin\theta = \sin(\theta + 360°)

— the periodicity that underlies waves, oscillations, and signal processing.

Coterminal reduction — bringing any angle into

[0°, 360°)

— is the standard first step before applying reference angles or reading signs in standard position.

Special Angles on the Unit Circle

Sixteen unit-circle positions have exact trigonometric values: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°. The scene shows them as a ring of clickable points.

The special angles, frozen

The multiples of 30° and 45° arranged around the circle — the positions whose exact values carry all of trigonometry.

The exact values come from just two triangles — the 30-60-90 and the 45-45-90 — reflected into all four quadrants, plus the axis points read directly from the circle's coordinates. That is why the whole ring reduces to the values of

30°

,

45°

, and

60°

with signs adjusted.

These sixteen positions are also the snap targets in the other scenes: dragging near one locks the handle onto the exact value.

Directed Angles

A directed angle carries a sign along with its magnitude: positive means counterclockwise rotation (blue), negative means clockwise (amber). Drag above or below the x-axis to switch direction.

Directed angles, frozen

The same magnitude swept two ways: counterclockwise counts positive, clockwise negative.

The panel verifies the two parity identities live as you drag:

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

(sine is odd) while

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

(cosine is even).

Direction matters wherever rotation is physical: phase in circuits and waves, bearings in navigation, and the argument of complex numbers all use signed angles.

Zero Angles

A zero angle has measure

0°

: both arms overlap along the same ray and no rotation has occurred. It is the starting state of the Angle Types scene and of every rotation in the tool.

Zero angle, frozen

Both rays coincide on the positive x-axis — no rotation at all.

Its trigonometric values anchor the unit circle:

\sin 0° = 0

,

\cos 0° = 1

,

\tan 0° = 0

.

The zero angle is coterminal with every multiple of

360°

— a full angle returns to exactly this position.

Acute Angles

An acute angle measures strictly between

0°

and

90°

— less than a quarter turn. The scene's Acute preset jumps to

45°

, drawn in blue.

Acute angle, frozen

A sweep caught between 0° and 90°, arc short of the perpendicular.

Acute angles live entirely in Quadrant I, where every trigonometric function is positive, and they are the raw material of right-triangle trigonometry: both non-right angles of any right triangle are acute.

Reference angles are acute by definition — which is why every other angle on the circle reduces to an acute one.

Right Angles

A right angle measures exactly

90°

— a quarter turn. The scene marks it with a small square at the vertex instead of the usual arc, the universal symbol for perpendicularity.

Right angle, frozen at 90°

The terminal ray exactly perpendicular, marked by the square corner.

At

90°

the values hit their quarter-turn landmarks:

\sin 90° = 1

,

\cos 90° = 0

, and

\tan 90°

is undefined because the tangent ratio divides by zero.

The right angle is the boundary between acute and obtuse, and the total that defines complementary pairs.

Obtuse Angles

An obtuse angle measures strictly between

90°

and

180°

— more than a quarter turn, less than a half. The scene's preset lands on

130°

, drawn in amber.

Obtuse angle, frozen

Wider than a right angle, narrower than a straight line.

An obtuse angle's terminal side points into Quadrant II, where sine stays positive while cosine and tangent turn negative. Its reference angle is

180° - \theta

.

Every obtuse angle is the supplement of an acute one: together they form a straight line, which is why a triangle can hold at most one obtuse angle.

Straight Angles

A straight angle measures exactly

180°

— a half turn. The two arms point in opposite directions and form a single straight line through the vertex, drawn in red by the scene.

Straight angle, frozen at 180°

The two rays form a single straight line through the vertex.

Its values mark the halfway point of the circle:

\sin 180° = 0

,

\cos 180° = -1

,

\tan 180° = 0

.

The straight angle defines the supplementary relationship and separates ordinary angles from reflex ones.

Reflex Angles

A reflex angle measures strictly between

180°

and

360°

— more than a half turn but less than a full one. The scene's preset shows

270°

, swept in purple the long way around.

Reflex angle, frozen

The rotation continues past 180°, sweeping the long way around.

Every arm position below the x-axis can be read two ways: as the reflex angle

\theta

or as its non-reflex partner

360° - \theta

on the other side. The tool always reports the counterclockwise sweep.

Reflex measures matter wherever full rotational position matters — bearings, phase, and rotations that pass the straight angle on their way toward a full angle.

Full Angles

A full angle measures exactly

360°

— one complete rotation. The scene draws it as an entire circle around the vertex: the moving arm has returned to the initial side.

Full angle, frozen at 360°

A complete turn: the terminal ray returns to its start.

Its values repeat the zero position exactly:

\sin 360° = 0

,

\cos 360° = 1

,

\tan 360° = 0

— the full angle is coterminal with the zero angle.

The full rotation is the period of sine and cosine: adding

360°

to any angle changes nothing about its trigonometry.

Complementary Angles

Two angles are complementary when their measures sum to

90°

: each is the other's complement,

90° - \theta

. The scene's Complementary tab constrains the blue and amber sectors to fill a quarter turn, as in the frozen state below (

35° + 55° = 90°

).

Complementary pair, frozen

Two angles stacked into one right angle — each is the other’s complement.

Complements are the "co" in cosine:

\cos\theta

literally means the sine of the complement,

\sin(90° - \theta)

. That is why sine and cosine swap values across a complementary pair, and why

45°

— its own complement — has them equal.

Geometrically, the two acute angles of every right triangle are complementary, filling the right angle's worth of turning that the triangle's third corner leaves over.

Supplementary Angles

Two angles are supplementary when their measures sum to

180°

: each is the other's supplement,

180° - \theta

. The scene's Supplementary tab stretches the constraint to a half turn — the frozen state shows

110° + 70° = 180°

.

Supplementary pair, frozen

Two angles completing a straight line — together exactly 180°.

Supplements share their sine and negate their cosine:

\sin(180° - \theta) = \sin\theta

while

\cos(180° - \theta) = -\cos\theta

.

Supplementary pairs appear wherever a straight angle is split: linear pairs at any line intersection — including the adjacent pairs of vertical angles — and co-interior angles between parallels.

Quadrant I: Function Signs

Quadrant I holds angles between

0°

and

90°

: both coordinates positive, so all six trigonometric functions are positive. The scene highlights the roman numeral I and shows the sign chart row for the current quadrant.

Standard position, Quadrant I

A terminal ray parked inside 0°–90°: the first-quadrant case of standard position.

The frozen state shows

50°

:

\sin

,

\cos

, and

\tan

all read

+

in the panel's sign cards.

First-quadrant values are the reference data for the entire circle — the reference angle in Quadrant I is the angle itself, no reduction needed.

Quadrant II: Function Signs

Quadrant II holds angles between

90°

and

180°

: x negative, y positive. Only sine (and cosecant) stays positive; cosine and tangent go negative.

Standard position, Quadrant II

The terminal ray between 90° and 180° — standard position in the second quadrant.

The frozen state shows

140°

: the sign cards read

\sin +

,

\cos -

,

\tan -

, and the sign-chart row for II is highlighted.

To evaluate here, combine the sign rule with the reference angle in Quadrant II:

\sin 140° = \sin 40°

while

\cos 140° = -\cos 40°

.

Quadrant III: Function Signs

Quadrant III holds angles between

180°

and

270°

: both coordinates negative. Sine and cosine are both negative, so their ratio — tangent (and cotangent) — is the survivor that stays positive.

Standard position, Quadrant III

Between 180° and 270°, the ray points into the third quadrant.

The frozen state shows

230°

: sign cards read

\sin -

,

\cos -

,

\tan +

.

Every Quadrant III angle is a Quadrant I angle shifted by

180°

— see the reference angle in Quadrant III — which is exactly why tangent's period is

180°

rather than

360°

.

Quadrant IV: Function Signs

Quadrant IV holds angles between

270°

and

360°

: x positive again, y still negative. Cosine (and secant) are positive; sine and tangent are negative.

Standard position, Quadrant IV

Between 270° and 360°, the ray closes the turn through the fourth quadrant.

The frozen state shows

320°

: sign cards read

\sin -

,

\cos +

,

\tan -

.

Quadrant IV angles mirror Quadrant I across the x-axis — the geometry behind cosine being even and sine odd, and the last stop before the rotation closes. Evaluate via the reference angle in Quadrant IV.

Reference Angle in Quadrant I

In Quadrant I the reference angle is the angle itself:

\theta_{ref} = \theta

. The terminal side already makes an acute angle with the positive x-axis, so no reduction is needed.

Reference angle, Quadrant I

In Quadrant I the angle IS its own reference angle — no folding needed.

The frozen state shows

50°

with its amber reference arc coinciding with the full blue rotation arc — they are the same angle here.

This identity case is what makes Quadrant I the reference data for the whole circle: all other quadrants' formulas fold their angles back onto this one. Signs are all positive here, per Quadrant I's sign rule.

Reference Angle in Quadrant II

In Quadrant II the terminal side is closest to the negative x-axis, so the reference angle measures back from

180°

:

\theta_{ref} = 180° - \theta

.

Reference angle, Quadrant II

The reference angle spans from the terminal ray back to the negative x-axis: 180° − θ.

The frozen state shows

140°

: the amber arc from the terminal side to the negative x-axis measures

180° - 140° = 40°

.

Worked through:

\sin 140° = \sin 40°

(sine positive in II) and

\cos 140° = -\cos 40°

(cosine negative), matching Quadrant II's sign rule.

Reference Angle in Quadrant III

In Quadrant III the terminal side has passed the negative x-axis, so the reference angle measures forward from

180°

:

\theta_{ref} = \theta - 180°

.

Reference angle, Quadrant III

Past 180°, the reference angle is measured onward from the negative x-axis: θ − 180°.

The frozen state shows

230°

: the amber arc from the negative x-axis to the terminal side measures

230° - 180° = 50°

.

Worked through:

\tan 230° = \tan 50°

(tangent positive in III) while

\sin 230° = -\sin 50°

and

\cos 230° = -\cos 50°

, per Quadrant III's sign rule.

Reference Angle in Quadrant IV

In Quadrant IV the terminal side approaches the positive x-axis from below, so the reference angle measures up to the full turn:

\theta_{ref} = 360° - \theta

.

Reference angle, Quadrant IV

Approaching the full turn, the reference angle is what remains: 360° − θ.

The frozen state shows

320°

: the amber arc from the terminal side up to

360°

measures

360° - 320° = 40°

.

Worked through:

\cos 320° = \cos 40°

(cosine positive in IV) while

\sin 320° = -\sin 40°

, per Quadrant IV's sign rule.

Special Angle: 0°

0°

sits on the positive x-axis — the starting point of the special-angle ring, with terminal point

(1, 0)

.

0°, frozen

Exact values at this position: sin = 0, cos = 1, tan = 0.

\sin 0° = 0 \qquad \cos 0° = 1 \qquad \tan 0° = 0

Cosine is at its maximum here and sine at zero. Every multiple of

360°

lands back on this point, which is why the ring treats

0°

and

360°

as the same position.

Special Angle: 30°

30°

is the smallest member of the 30-60-90 family, one-twelfth of a turn,

\pi/6

in radians.

30°, frozen

Exact values at this position: sin = 1/2, cos = √3/2, tan = 1/√3.

\sin 30° = \frac{1}{2} \qquad \cos 30° = \frac{\sqrt{3}}{2} \qquad \tan 30° = \frac{\sqrt{3}}{3}

The half comes straight from the 30-60-90 triangle, where the side opposite

30°

is half the hypotenuse. Its reflections at

150°

,

210°

, and

330°

reuse these magnitudes with quadrant signs.

Special Angle: 45°

45°

bisects the right angle along the diagonal

y = x

,

\pi/4

in radians. Its sine and cosine are equal.

45°, frozen

Exact values at this position: sin = √2/2, cos = √2/2, tan = 1.

\sin 45° = \frac{\sqrt{2}}{2} \qquad \cos 45° = \frac{\sqrt{2}}{2} \qquad \tan 45° = 1

The values come from the isosceles 45-45-90 triangle with legs

1

and hypotenuse

\sqrt{2}

. Its reflections live at

135°

,

225°

, and

315°

.

Special Angle: 60°

60°

is the larger acute angle of the 30-60-90 triangle and the interior angle of the equilateral triangle,

\pi/3

in radians.

60°, frozen

Exact values at this position: sin = √3/2, cos = 1/2, tan = √3.

\sin 60° = \frac{\sqrt{3}}{2} \qquad \cos 60° = \frac{1}{2} \qquad \tan 60° = \sqrt{3}

Sine and cosine swap compared to

30°

because the two angles are complementary. Its reflections live at

120°

,

240°

, and

300°

.

Special Angle: 90°

90°

points straight up the positive y-axis — the quarter turn,

\pi/2

in radians, terminal point

(0, 1)

.

90°, frozen

Exact values at this position: sin = 1, cos = 0, tan = undefined.

\sin 90° = 1 \qquad \cos 90° = 0 \qquad \tan 90° \text{ undefined}

Sine peaks at its maximum while cosine crosses zero, so the tangent ratio divides by zero — the first of the two undefined points on the ring.

Special Angle: 120°

120°

lies in Quadrant II,

30°

past vertical — the supplement of

60°

and the hexagon's interior angle,

2\pi/3

in radians.

120°, frozen

Exact values at this position: sin = √3/2, cos = −1/2, tan = −√3.

\sin 120° = \frac{\sqrt{3}}{2} \qquad \cos 120° = -\frac{1}{2} \qquad \tan 120° = -\sqrt{3}

The magnitudes are exactly the

60°

values; only the signs change to Quadrant II's pattern of sine positive, cosine and tangent negative.

Special Angle: 135°

135°

is the Quadrant II diagonal along

y = -x

, supplement of

45°

,

3\pi/4

in radians.

135°, frozen

Exact values at this position: sin = √2/2, cos = −√2/2, tan = −1.

\sin 135° = \frac{\sqrt{2}}{2} \qquad \cos 135° = -\frac{\sqrt{2}}{2} \qquad \tan 135° = -1

With reference angle

45°

, sine and cosine share one magnitude and differ only in sign — forcing the tangent to exactly

-1

.

Special Angle: 150°

150°

sits

30°

short of the straight line, in Quadrant II — the supplement of

30°

,

5\pi/6

in radians.

150°, frozen

Exact values at this position: sin = 1/2, cos = −√3/2, tan = −1/√3.

\sin 150° = \frac{1}{2} \qquad \cos 150° = -\frac{\sqrt{3}}{2} \qquad \tan 150° = -\frac{\sqrt{3}}{3}

The classic supplement demonstration:

\sin 150° = \sin 30°

exactly, while the cosine flips sign.

Special Angle: 180°

180°

points along the negative x-axis — the half turn,

\pi

radians, terminal point

(-1, 0)

.

180°, frozen

Exact values at this position: sin = 0, cos = −1, tan = 0.

\sin 180° = 0 \qquad \cos 180° = -1 \qquad \tan 180° = 0

Cosine bottoms out at its minimum while sine returns to zero. This point is the pivot of the supplement relationship

\theta \mapsto 180° - \theta

.

Special Angle: 210°

210°

lies

30°

past the straight line, in Quadrant III —

7\pi/6

in radians, reference angle

30°

.

210°, frozen

Exact values at this position: sin = −1/2, cos = −√3/2, tan = 1/√3.

\sin 210° = -\frac{1}{2} \qquad \cos 210° = -\frac{\sqrt{3}}{2} \qquad \tan 210° = \frac{\sqrt{3}}{3}

Both coordinates go negative here, so their ratio — the tangent — turns positive again, matching

\tan 30°

exactly.

Special Angle: 225°

225°

is the Quadrant III diagonal, opposite

45°

through the origin —

5\pi/4

in radians.

225°, frozen

Exact values at this position: sin = −√2/2, cos = −√2/2, tan = 1.

\sin 225° = -\frac{\sqrt{2}}{2} \qquad \cos 225° = -\frac{\sqrt{2}}{2} \qquad \tan 225° = 1

Equal negative coordinates give a tangent of exactly

1

again:

225° = 45° + 180°

, tangent's period made visible.

Special Angle: 240°

240°

lies

60°

past the straight line in Quadrant III, diametrically opposite

60°

—

4\pi/3

in radians.

240°, frozen

Exact values at this position: sin = −√3/2, cos = −1/2, tan = √3.

\sin 240° = -\frac{\sqrt{3}}{2} \qquad \cos 240° = -\frac{1}{2} \qquad \tan 240° = \sqrt{3}

The 30-60-90 magnitudes reappear with both signs flipped, leaving the tangent at a positive

\sqrt{3}

.

Special Angle: 270°

270°

points straight down the negative y-axis — the three-quarter turn,

3\pi/2

radians, terminal point

(0, -1)

.

270°, frozen

Exact values at this position: sin = −1, cos = 0, tan = undefined.

\sin 270° = -1 \qquad \cos 270° = 0 \qquad \tan 270° \text{ undefined}

Sine bottoms out at its minimum while cosine crosses zero again — the second and last undefined point for the tangent on the ring.

Special Angle: 300°

300°

lies in Quadrant IV,

60°

short of the full turn and coterminal with

-60°

—

5\pi/3

in radians.

300°, frozen

Exact values at this position: sin = −√3/2, cos = 1/2, tan = −√3.

\sin 300° = -\frac{\sqrt{3}}{2} \qquad \cos 300° = \frac{1}{2} \qquad \tan 300° = -\sqrt{3}

Cosine turns positive again in Quadrant IV while sine stays negative — the mirror image of

60°

across the x-axis.

Special Angle: 315°

315°

is the Quadrant IV diagonal,

45°

short of the full turn and coterminal with

-45°

—

7\pi/4

in radians.

315°, frozen

Exact values at this position: sin = −√2/2, cos = √2/2, tan = −1.

\sin 315° = -\frac{\sqrt{2}}{2} \qquad \cos 315° = \frac{\sqrt{2}}{2} \qquad \tan 315° = -1

The

45°

magnitudes return with opposite signs on sine and cosine, making the tangent exactly

-1

.

Special Angle: 330°

330°

sits

30°

short of closing the rotation, in Quadrant IV, coterminal with

-30°

—

11\pi/6

in radians.

330°, frozen

Exact values at this position: sin = −1/2, cos = √3/2, tan = −1/√3.

\sin 330° = -\frac{1}{2} \qquad \cos 330° = \frac{\sqrt{3}}{2} \qquad \tan 330° = -\frac{\sqrt{3}}{3}

The familiar

30°

magnitudes make their final appearance before the ring closes, with sine negative below the axis.

Positive Angles

A positive angle rotates counterclockwise from the initial side — the mathematical convention for the positive direction, drawn blue in the scene.

Positive angle, frozen

A counterclockwise sweep — the mathematically positive direction of rotation.

All the standard machinery — quadrants, reference angles, special values — is defined for counterclockwise sweeps first; clockwise angles are handled by symmetry.

Every positive angle has a negative twin pointing to the same terminal side:

+45°

and

-315°

are coterminal.

Negative Angles

A negative angle rotates clockwise from the initial side, drawn amber in the scene. Drag below the x-axis in the Directed Angles view to produce one.

Negative angle, frozen

The same start, swept clockwise: the angle carries a minus sign.

Negation is where the parity of the functions shows:

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

makes sine an odd function, while

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

makes cosine even — both verified live in the panel.

Adding

360°

converts any negative angle to its positive coterminal twin:

-45°

is the same terminal side as

315°

.