Why does sine of an angle change sign between quadrants?

Sine is defined as y divided by r. The radius r is always positive, so the sign of sine matches the sign of the y-coordinate. Because y is positive in quadrants I and II and negative in quadrants III and IV, sine flips sign across the x-axis.

What is the ASTC rule?

ASTC summarizes which trig functions are positive in each quadrant: All in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4. The common memory phrase All Students Take Calculus walks the quadrants counterclockwise starting from Q1.

Do reciprocal functions have the same signs as their counterparts?

Yes. Cosecant matches sine, secant matches cosine, and cotangent matches tangent in every quadrant. The reciprocal of a positive number is positive and the reciprocal of a negative number is negative, so the signs always agree.

How do I select a quadrant in the tool?

Click any of the four quadrant buttons at the top, or press the number keys 1, 2, 3, or 4 on your keyboard. The unit circle highlights the chosen region and the explanation panel updates instantly.

Why is the radius always positive in trigonometry?

The radius r is the distance from the origin to a point on the unit circle, and distance is non-negative by definition. This means the sign of every trig ratio depends only on the signs of the x and y coordinates, never on r.

Trig Function Signs by Quadrant: ASTC Visualizer

Selecting a Quadrant

Four large buttons at the top represent quadrants I, II, III, and IV. Each button shows the angle range in both degrees and radians.

Two ways to select:
• Click any quadrant button.
• Press the keys 1, 2, 3, or 4 for the matching quadrant.

When you select a quadrant:
• Its color highlights both the button and the corresponding region of the unit circle.
• A point and radius appear inside that region of the circle.
• The coordinate panel updates with the matching x-sign and y-sign.
• The explanation panel and function summary refresh with values for the new quadrant.

Choosing a Function

Six buttons let you switch between the trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent.

What changes when you switch:
• The header of the Explanation Panel updates to show

f(\theta)

in the selected quadrant.
• A large

+

or

-

symbol displays the function's sign.
• The reasoning text below derives the sign from the coordinate signs and the function's definition.

The same six functions also appear in the Function Summary grid below the panel, where you can see every sign for the current quadrant at once.

Reading the Unit Circle Diagram

The interactive unit circle anchors the entire tool. The selected quadrant is shaded in its color, and a colored radius runs from the origin to a sample point inside it.

What the diagram shows:
• Solid axes labeled

+x

,

-x

,

+y

,

-y

.
• Roman numeral labels I, II, III, IV in each region.
• A point

(x, y)

on the circle with dashed lines dropping to the axes.
• A radius labeled

r

, always positive, from origin to point.

This setup makes the connection between coordinate signs and function signs visible:

x

and

y

change sign across quadrants, but

r

never does.

Reading the Coordinate Panel

Below the unit circle, the Coordinate Panel lists the algebraic facts that determine every sign in the current quadrant.

Fields shown:
• x-coordinate — colored green for

+

or red for

-

.
• y-coordinate — colored green for

+

or red for

-

.
• radius (r) — always positive.
• Reference angle — the angle range in degrees and radians.

Memorizing the four x and y sign patterns is the single most important step in mastering trig signs. The panel surfaces them so you can see all four at once by clicking through the quadrants.

Reading the Explanation Panel

The Explanation Panel combines the current function and quadrant into a single, color-coded summary.

Three pieces of information:
• Header — names the function and quadrant, colored to match the selected region.
• Sign — a large

+

or

-

symbol gives the answer at a glance.
• Mathematical Reasoning — a sentence derives the sign from the function's definition (e.g.,

\tan\theta = y/x

) and the coordinate signs of the quadrant.

Toggle the Hide / Show Explanations button at the top of the page to remove the reasoning text and use the tool as a quick sign reference.

Using the Function Summary Grid

The Function Summary at the bottom shows all six functions and their signs for the current quadrant in a compact grid.

Why it is useful:
• Verify the ASTC pattern at a glance — for example, in Q3 only

\tan

and

\cot

are positive.
• Confirm reciprocal pairs share signs (

\sin

matches

\csc

,

\cos

matches

\sec

,

\tan

matches

\cot

).
• Click any cell to make that function the active selection in the explanation panel above.

Cycling through quadrants while keeping an eye on this grid is the fastest way to internalize the full sign chart.

Why Function Signs Depend on Quadrant

The trigonometric functions are defined as ratios involving the coordinates of a point on the unit circle:

\sin\theta = \frac{y}{r}, \quad \cos\theta = \frac{x}{r}, \quad \tan\theta = \frac{y}{x}

Because

r > 0

everywhere, the signs of these ratios depend entirely on the signs of

x

and

y

. Since

x

and

y

flip sign as the terminal ray crosses an axis, each function takes on a predictable sign in each quadrant.

For deeper coverage of these definitions, see the trigonometric functions theory page.

The ASTC Mnemonic

The ASTC rule (also called CAST) summarizes which functions are positive in each quadrant:

• Q1 — All six functions are positive.
• Q2 — only Sine (and its reciprocal

\csc

) are positive.
• Q3 — only Tangent (and its reciprocal

\cot

) are positive.
• Q4 — only Cosine (and its reciprocal

\sec

) are positive.

Common phrase: All Students Take Calculus. Walking quadrants counterclockwise from Q1, the first letter of each word names the positive function family.

For full context, see the ASTC rule page.

Reciprocal Functions Share Signs

A reciprocal of a number always has the same sign as the number itself, since

1/(+x)

is positive and

1/(-x)

is negative.

Applied to trigonometry:
•

\csc\theta = 1/\sin\theta

shares its sign with

\sin\theta

.
•

\sec\theta = 1/\cos\theta

shares its sign with

\cos\theta

.
•

\cot\theta = 1/\tan\theta

shares its sign with

\tan\theta

.

This means you only need to memorize the signs of three functions (

\sin

,

\cos

,

\tan

). The other three follow automatically. For more, see the reciprocal identities page.

Related Concepts and Tools

Continue exploring with these connected resources:

• Unit Circle — the geometric setup behind every quadrant rule.
• Angle Explorer — visualize any angle, its quadrant, and its reference angle.
• Trigonometric Functions Graphs — see how signs translate into the shapes of

\sin

,

\cos

, and

\tan

curves.
• Reference Angle Calculator — reduce any angle to a first-quadrant equivalent.
• Trigonometric Identities — Pythagorean, reciprocal, and quotient identities that interact with these signs.
• Special Angles Table — exact values across all four quadrants.

Coordinate Signs in Quadrant I

Quadrant I spans

0°

–

90°

(

0

to

\pi/2

in radians): the x-coordinate is positive, the y-coordinate is positive, and the radius

r

is always positive. Every function sign in this quadrant follows from those three facts. Its ASTC letter is A — all six functions positive.

Quadrant I, frozen

The reference point sits where x is + and y is + — the two coordinate signs from which every function sign in this quadrant follows.

Function by function here: sine +, cosine +, tangent +, cosecant +, secant +, cotangent +.

Coordinate Signs in Quadrant II

Quadrant II spans

90°

–

180°

(

\pi/2

to

\pi

in radians): the x-coordinate is negative, the y-coordinate is positive, and the radius

r

is always positive. Every function sign in this quadrant follows from those three facts. Its ASTC letter is S — only sine and its reciprocal positive.

Quadrant II, frozen

The reference point sits where x is − and y is + — the two coordinate signs from which every function sign in this quadrant follows.

Function by function here: sine +, cosine −, tangent −, cosecant +, secant −, cotangent −.

Coordinate Signs in Quadrant III

Quadrant III spans

180°

–

270°

(

\pi

to

3\pi/2

in radians): the x-coordinate is negative, the y-coordinate is negative, and the radius

r

is always positive. Every function sign in this quadrant follows from those three facts. Its ASTC letter is T — only tangent and its reciprocal positive.

Quadrant III, frozen

The reference point sits where x is − and y is − — the two coordinate signs from which every function sign in this quadrant follows.

Function by function here: sine −, cosine −, tangent +, cosecant −, secant −, cotangent +.

Coordinate Signs in Quadrant IV

Quadrant IV spans

270°

–

360°

(

3\pi/2

to

2\pi

in radians): the x-coordinate is positive, the y-coordinate is negative, and the radius

r

is always positive. Every function sign in this quadrant follows from those three facts. Its ASTC letter is C — only cosine and its reciprocal positive.

Quadrant IV, frozen

The reference point sits where x is + and y is − — the two coordinate signs from which every function sign in this quadrant follows.

Function by function here: sine −, cosine +, tangent −, cosecant −, secant +, cotangent −.

Sine in Quadrant I

In Quadrant I (

0°

–

90°

), sine is positive:

\sin\theta = y/r

, and the y-coordinate is positive there while

r

stays positive.

Sine in Quadrant I: +

The frozen scene marks the positive value: sine is the y-coordinate, and in Quadrant I that makes it positive.

At the sample angle the diagram freezes,

\sin 45° = \frac{\sqrt{2}}{2}

(\approx 0.707). Its reciprocal partner, cosecant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant I.

Cosine in Quadrant I

In Quadrant I (

0°

–

90°

), cosine is positive:

\cos\theta = x/r

, and the x-coordinate is positive there while

r

stays positive.

Cosine in Quadrant I: +

The frozen scene marks the positive value: cosine is the x-coordinate, and in Quadrant I that makes it positive.

At the sample angle the diagram freezes,

\cos 45° = \frac{\sqrt{2}}{2}

(\approx 0.707). Its reciprocal partner, secant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant I.

Tangent in Quadrant I

In Quadrant I (

0°

–

90°

), tangent is positive:

\tan\theta = y/x

, and y is positive and x is positive, so their ratio is positive.

Tangent in Quadrant I: +

The frozen scene marks the positive value: tangent divides y by x, and in Quadrant I that makes it positive.

At the sample angle the diagram freezes,

\tan 45° = 1

. Its reciprocal partner, cotangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant I.

Cosecant in Quadrant I

In Quadrant I (

0°

–

90°

), cosecant is positive:

\csc\theta = r/y

, and it inherits sine's sign — y is positive,

r

positive.

Cosecant in Quadrant I: +

The frozen scene marks the positive value: cosecant inherits sine’s sign, and in Quadrant I that makes it positive.

At the sample angle the diagram freezes,

\csc 45° = \sqrt{2}

(\approx 1.414). Its reciprocal partner, sine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant I.

Secant in Quadrant I

In Quadrant I (

0°

–

90°

), secant is positive:

\sec\theta = r/x

, and it inherits cosine's sign — x is positive,

r

positive.

Secant in Quadrant I: +

The frozen scene marks the positive value: secant inherits cosine’s sign, and in Quadrant I that makes it positive.

At the sample angle the diagram freezes,

\sec 45° = \sqrt{2}

(\approx 1.414). Its reciprocal partner, cosine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant I.

Cotangent in Quadrant I

In Quadrant I (

0°

–

90°

), cotangent is positive:

\cot\theta = x/y

, and y is positive and x is positive, so their ratio is positive.

Cotangent in Quadrant I: +

The frozen scene marks the positive value: cotangent inherits tangent’s sign, and in Quadrant I that makes it positive.

At the sample angle the diagram freezes,

\cot 45° = 1

. Its reciprocal partner, tangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant I.

Sine in Quadrant II

In Quadrant II (

90°

–

180°

), sine is positive:

\sin\theta = y/r

, and the y-coordinate is positive there while

r

stays positive.

Sine in Quadrant II: +

The frozen scene marks the positive value: sine is the y-coordinate, and in Quadrant II that makes it positive.

At the sample angle the diagram freezes,

\sin 135° = \frac{\sqrt{2}}{2}

(\approx 0.707). Its reciprocal partner, cosecant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant II.

Cosine in Quadrant II

In Quadrant II (

90°

–

180°

), cosine is negative:

\cos\theta = x/r

, and the x-coordinate is negative there while

r

stays positive.

Cosine in Quadrant II: −

The frozen scene marks the negative value: cosine is the x-coordinate, and in Quadrant II that makes it negative.

At the sample angle the diagram freezes,

\cos 135° = -\frac{\sqrt{2}}{2}

(\approx -0.707). Its reciprocal partner, secant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant II.

Tangent in Quadrant II

In Quadrant II (

90°

–

180°

), tangent is negative:

\tan\theta = y/x

, and y is positive and x is negative, so their ratio is negative.

Tangent in Quadrant II: −

The frozen scene marks the negative value: tangent divides y by x, and in Quadrant II that makes it negative.

At the sample angle the diagram freezes,

\tan 135° = -1

. Its reciprocal partner, cotangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant II.

Cosecant in Quadrant II

In Quadrant II (

90°

–

180°

), cosecant is positive:

\csc\theta = r/y

, and it inherits sine's sign — y is positive,

r

positive.

Cosecant in Quadrant II: +

The frozen scene marks the positive value: cosecant inherits sine’s sign, and in Quadrant II that makes it positive.

At the sample angle the diagram freezes,

\csc 135° = \sqrt{2}

(\approx 1.414). Its reciprocal partner, sine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant II.

Secant in Quadrant II

In Quadrant II (

90°

–

180°

), secant is negative:

\sec\theta = r/x

, and it inherits cosine's sign — x is negative,

r

positive.

Secant in Quadrant II: −

The frozen scene marks the negative value: secant inherits cosine’s sign, and in Quadrant II that makes it negative.

At the sample angle the diagram freezes,

\sec 135° = -\sqrt{2}

(\approx -1.414). Its reciprocal partner, cosine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant II.

Cotangent in Quadrant II

In Quadrant II (

90°

–

180°

), cotangent is negative:

\cot\theta = x/y

, and y is positive and x is negative, so their ratio is negative.

Cotangent in Quadrant II: −

The frozen scene marks the negative value: cotangent inherits tangent’s sign, and in Quadrant II that makes it negative.

At the sample angle the diagram freezes,

\cot 135° = -1

. Its reciprocal partner, tangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant II.

Sine in Quadrant III

In Quadrant III (

180°

–

270°

), sine is negative:

\sin\theta = y/r

, and the y-coordinate is negative there while

r

stays positive.

Sine in Quadrant III: −

The frozen scene marks the negative value: sine is the y-coordinate, and in Quadrant III that makes it negative.

At the sample angle the diagram freezes,

\sin 225° = -\frac{\sqrt{2}}{2}

(\approx -0.707). Its reciprocal partner, cosecant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant III.

Cosine in Quadrant III

In Quadrant III (

180°

–

270°

), cosine is negative:

\cos\theta = x/r

, and the x-coordinate is negative there while

r

stays positive.

Cosine in Quadrant III: −

The frozen scene marks the negative value: cosine is the x-coordinate, and in Quadrant III that makes it negative.

At the sample angle the diagram freezes,

\cos 225° = -\frac{\sqrt{2}}{2}

(\approx -0.707). Its reciprocal partner, secant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant III.

Tangent in Quadrant III

In Quadrant III (

180°

–

270°

), tangent is positive:

\tan\theta = y/x

, and y is negative and x is negative, so their ratio is positive.

Tangent in Quadrant III: +

The frozen scene marks the positive value: tangent divides y by x, and in Quadrant III that makes it positive.

At the sample angle the diagram freezes,

\tan 225° = 1

. Its reciprocal partner, cotangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant III.

Cosecant in Quadrant III

In Quadrant III (

180°

–

270°

), cosecant is negative:

\csc\theta = r/y

, and it inherits sine's sign — y is negative,

r

positive.

Cosecant in Quadrant III: −

The frozen scene marks the negative value: cosecant inherits sine’s sign, and in Quadrant III that makes it negative.

At the sample angle the diagram freezes,

\csc 225° = -\sqrt{2}

(\approx -1.414). Its reciprocal partner, sine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant III.

Secant in Quadrant III

In Quadrant III (

180°

–

270°

), secant is negative:

\sec\theta = r/x

, and it inherits cosine's sign — x is negative,

r

positive.

Secant in Quadrant III: −

The frozen scene marks the negative value: secant inherits cosine’s sign, and in Quadrant III that makes it negative.

At the sample angle the diagram freezes,

\sec 225° = -\sqrt{2}

(\approx -1.414). Its reciprocal partner, cosine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant III.

Cotangent in Quadrant III

In Quadrant III (

180°

–

270°

), cotangent is positive:

\cot\theta = x/y

, and y is negative and x is negative, so their ratio is positive.

Cotangent in Quadrant III: +

The frozen scene marks the positive value: cotangent inherits tangent’s sign, and in Quadrant III that makes it positive.

At the sample angle the diagram freezes,

\cot 225° = 1

. Its reciprocal partner, tangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant III.

Sine in Quadrant IV

In Quadrant IV (

270°

–

360°

), sine is negative:

\sin\theta = y/r

, and the y-coordinate is negative there while

r

stays positive.

Sine in Quadrant IV: −

The frozen scene marks the negative value: sine is the y-coordinate, and in Quadrant IV that makes it negative.

At the sample angle the diagram freezes,

\sin 315° = -\frac{\sqrt{2}}{2}

(\approx -0.707). Its reciprocal partner, cosecant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant IV.

Cosine in Quadrant IV

In Quadrant IV (

270°

–

360°

), cosine is positive:

\cos\theta = x/r

, and the x-coordinate is positive there while

r

stays positive.

Cosine in Quadrant IV: +

The frozen scene marks the positive value: cosine is the x-coordinate, and in Quadrant IV that makes it positive.

At the sample angle the diagram freezes,

\cos 315° = \frac{\sqrt{2}}{2}

(\approx 0.707). Its reciprocal partner, secant, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant IV.

Tangent in Quadrant IV

In Quadrant IV (

270°

–

360°

), tangent is negative:

\tan\theta = y/x

, and y is negative and x is positive, so their ratio is negative.

Tangent in Quadrant IV: −

The frozen scene marks the negative value: tangent divides y by x, and in Quadrant IV that makes it negative.

At the sample angle the diagram freezes,

\tan 315° = -1

. Its reciprocal partner, cotangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant IV.

Cosecant in Quadrant IV

In Quadrant IV (

270°

–

360°

), cosecant is negative:

\csc\theta = r/y

, and it inherits sine's sign — y is negative,

r

positive.

Cosecant in Quadrant IV: −

The frozen scene marks the negative value: cosecant inherits sine’s sign, and in Quadrant IV that makes it negative.

At the sample angle the diagram freezes,

\csc 315° = -\sqrt{2}

(\approx -1.414). Its reciprocal partner, sine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant IV.

Secant in Quadrant IV

In Quadrant IV (

270°

–

360°

), secant is positive:

\sec\theta = r/x

, and it inherits cosine's sign — x is positive,

r

positive.

Secant in Quadrant IV: +

The frozen scene marks the positive value: secant inherits cosine’s sign, and in Quadrant IV that makes it positive.

At the sample angle the diagram freezes,

\sec 315° = \sqrt{2}

(\approx 1.414). Its reciprocal partner, cosine, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant IV.

Cotangent in Quadrant IV

In Quadrant IV (

270°

–

360°

), cotangent is negative:

\cot\theta = x/y

, and y is negative and x is positive, so their ratio is negative.

Cotangent in Quadrant IV: −

The frozen scene marks the negative value: cotangent inherits tangent’s sign, and in Quadrant IV that makes it negative.

At the sample angle the diagram freezes,

\cot 315° = -1

. Its reciprocal partner, tangent, carries the same sign here; the underlying coordinate facts are in coordinate signs in Quadrant IV.

Trigonometric Functions Signs by Quadrants

Quadrant 1 Coordinates

sin(θ) in Quadrant 1

All Functions in Quadrant 1

Key Concepts

Selecting a Quadrant

Choosing a Function

Reading the Unit Circle Diagram

Reading the Coordinate Panel

Reading the Explanation Panel

Using the Function Summary Grid

Why Function Signs Depend on Quadrant

The ASTC Mnemonic

Reciprocal Functions Share Signs

Related Concepts and Tools

Coordinate Signs in Quadrant I

Coordinate Signs in Quadrant II

Coordinate Signs in Quadrant III

Coordinate Signs in Quadrant IV

Sine in Quadrant I

Cosine in Quadrant I

Tangent in Quadrant I

Cosecant in Quadrant I

Secant in Quadrant I

Cotangent in Quadrant I

Sine in Quadrant II

Cosine in Quadrant II

Tangent in Quadrant II

Cosecant in Quadrant II

Secant in Quadrant II

Cotangent in Quadrant II

Sine in Quadrant III

Cosine in Quadrant III

Tangent in Quadrant III

Cosecant in Quadrant III

Secant in Quadrant III

Cotangent in Quadrant III

Sine in Quadrant IV

Cosine in Quadrant IV

Tangent in Quadrant IV

Cosecant in Quadrant IV

Secant in Quadrant IV

Cotangent in Quadrant IV