Trigonometric Signs by Quadrant: The Unit-Circle Method

You know that sin 30° is 1/2. Then the question asks for sin 210°, and you pause. Is it positive because the angle is positive? Does crossing 180° change every sign? Which part of the quadrant shortcut should you use?

There is a calmer way to decide. Put the angle on a circle and look at where its endpoint sits. Left or right tells you the sign of cosine. Above or below tells you the sign of sine. Division gives you tangent, and reciprocals give you the remaining three functions.

Once you see that connection, the sign chart becomes something you can rebuild in a few seconds.

This guide uses θ for an angle, π for pi, and sqrt for the nonnegative square root. A degree symbol means degrees; angles written using π are in radians.

Draw the circle before learning the chart

Draw a horizontal x-axis and a vertical y-axis. Their intersection is the origin, O. Now draw a circle of radius 1 centred at O. That is the unit circle.

Start a ray along the positive x-axis, pointing right. Rotate it through an angle θ. Its final position is called the terminal side. Let P(x, y) be the point where that side meets the circle.

The definitions are:

cos θ = x and sin θ = y.

Positive rotation is anticlockwise; negative rotation is clockwise. These conventions and the coordinate definitions are given in NCERT’s Trigonometric Functions chapter, Sections 3.2 and 3.3.

Why do these definitions agree with the right triangles you already know? In the first quadrant, drop a perpendicular from P to the x-axis. The hypotenuse is the circle’s radius, 1. The horizontal side has length x and the vertical side has length y. So adjacent/hypotenuse is x/1, and opposite/hypotenuse is y/1.

Outside that quadrant, coordinates let us keep track of direction. The distance to the left is still a positive distance, but its x-coordinate is negative. This is why using only positive triangle lengths can lose a sign.

Build the quadrant signs from coordinates

Quadrants are numbered anticlockwise, starting in the upper-right region. The axes themselves are boundaries, so an angle on an axis does not lie inside a quadrant.

Quadrant	Angle within one turn	Position of P	(x sign, y sign)
I	0° < θ < 90°	Right and above	(+, +)
II	90° < θ < 180°	Left and above	(-, +)
III	180° < θ < 270°	Left and below	(-, -)
IV	270° < θ < 360°	Right and below	(+, -)

In radians, the boundaries are 0, π/2, π, 3π/2, and 2π.

Now read the coordinates. Sine is positive above the horizontal axis, so it is positive in I and II. Cosine is positive to the right of the vertical axis, so it is positive in I and IV.

For tangent, use tan θ = y/x, provided x is not zero. Two coordinates with the same sign give a positive quotient. Coordinates with opposite signs give a negative quotient. That makes tangent positive in I and III.

Here is the full chart, grouped into pairs so you only need three columns:

Quadrant	sin and cosec	cos and sec	tan and cot
I	Positive	Positive	Positive
II	Positive	Negative	Negative
III	Negative	Negative	Positive
IV	Negative	Positive	Negative

Some students remember the positive families as All, Sine, Tangent, Cosine while moving through I, II, III, IV. That shortcut is useful once you know why it works. The coordinate picture also handles cases that a mnemonic cannot settle by itself.

Why do reciprocal functions have the same sign?

The remaining definitions are:

• cosec θ = 1/y, when y is nonzero.
• sec θ = 1/x, when x is nonzero.
• cot θ = x/y, when y is nonzero.

A reciprocal preserves the sign of a nonzero number. For instance, the reciprocal of -2/5 is -5/2. So cosecant follows sine, and secant follows cosine. In an open quadrant, cotangent and tangent also have the same sign.

OpenStax’s lesson on the other trigonometric functions develops these quotient and reciprocal definitions. Notice the restrictions: dividing by zero is not allowed.

A four-step routine for any angle

1. Remove complete turns. Add or subtract multiples of 360°, or 2π, to locate an equivalent angle within one turn.
2. Check for an axis. If the endpoint lies on an axis, use its coordinates directly.
3. Find the quadrant. Read the signs of x and y, then use the required function’s definition.
4. Find the magnitude if needed. Use a reference angle or an identity, keeping the sign you already decided.

A reference angle is the acute angle between a non-axis terminal side and the x-axis. It helps with the size of the answer; the original endpoint determines its sign. OpenStax’s unit-circle lesson explains this separation.

For an angle reduced to one turn, the reference angle is θ in I, 180° - θ in II, θ - 180° in III, and 360° - θ in IV. In radians, replace 180° with π and 360° with 2π.

Worked examples: make the sign decision visible

1. Find the signs of all six functions at 142°

142° is between 90° and 180°, so the terminal point is in quadrant II. It sits left of the origin and above the x-axis: x < 0, y > 0.

Therefore sine and cosecant are positive. Cosine and secant are negative. Tangent and cotangent are negative because they divide coordinates with opposite signs.

We do not need the exact value of any function. A sign question can be finished without a calculator or a special angle.

2. Find sin 240°, cos 240°, and tan 240°

240° is in quadrant III, where both coordinates are negative. Its reference angle is 240° - 180° = 60°.

The familiar magnitudes are sin 60° = sqrt(3)/2 and cos 60° = 1/2. Attach the signs from quadrant III:

sin 240° = -sqrt(3)/2.

cos 240° = -1/2.

Then divide:

tan 240° = [-sqrt(3)/2] / [-1/2] = sqrt(3).

This example is a good reminder that a positive angle can have negative sine and cosine, while their quotient is positive.

3. Find sec(-300°)

Add one complete turn:

-300° + 360° = 60°.

The two angles have the same terminal side. At 60°, cosine is 1/2, so:

sec(-300°) = 1 / (1/2) = 2.

Do not give the answer a minus sign just because the angle begins with one. A negative angle tells you the direction of rotation, not the sign of every trigonometric value.

4. Find cosec 765°

Subtract two complete turns:

765° - 720° = 45°.

The endpoint is in quadrant I. Sine is positive, and sin 45° = 1/sqrt(2). Therefore:

cosec 765° = sqrt(2).

Complete turns change how far you travelled, but they do not change where you finished.

5. Find cos(17π/6)

Subtract 2π, which is 12π/6:

17π/6 - 12π/6 = 5π/6.

This is in quadrant II. The reference angle is π - 5π/6 = π/6. Cosine is negative there, so:

cos(17π/6) = -sqrt(3)/2.

Keep a common denominator when reducing radian angles. It makes the full-turn subtraction easier to see.

6. Given sin θ = 8/17 in quadrant II, find the other five functions

The identity sin²θ + cos²θ = 1 gives:

cos²θ = 1 - 64/289 = 225/289.

Algebra alone gives two possibilities, 15/17 and -15/17. The quadrant chooses the negative one because the point is left of the origin.

cos θ = -15/17.

Now calculate the rest:

• tan θ = (8/17) / (-15/17) = -8/15.
• cosec θ = 17/8.
• sec θ = -17/15.
• cot θ = -15/8.

Check the families: sine and cosecant are positive, while the other four are negative. Every result fits the point’s position.

7. Given tan θ = 5/12 and sin θ < 0, find sin θ and cos θ

A positive tangent permits quadrant I or III. A negative sine permits III or IV. The only common choice is quadrant III.

Use a representative point (-12, -5) on the terminal side. Its distance from the origin is sqrt(144 + 25) = 13. Therefore:

sin θ = -5/13 and cos θ = -12/13.

The ratio (-5)/(-12) is indeed 5/12. This check matters: drawing a triangle with positive lengths 5, 12, and 13 gives the magnitudes, but the quadrant must still supply the coordinate signs.

8. Determine the sign of sin θ cos θ when 270° < θ < 360°

The angle is in quadrant IV. Sine is negative and cosine is positive, so their product is negative.

No exact angle is needed. The same idea works for longer expressions: decide each factor’s sign, then multiply or divide the signs. If there is a denominator, also check that it is not zero.

Axis angles: zero and undefined are different answers

At the four axis positions, the unit-circle points are (1, 0), (0, 1), (-1, 0), and (0, -1). Substituting these into the definitions is safer than extending a quadrant mnemonic to a boundary.

Angle	(x, y)	sin θ	cos θ	tan θ
0° or 360°	(1, 0)	0	1	0
90°	(0, 1)	1	0	Undefined
180°	(-1, 0)	0	-1	0
270°	(0, -1)	-1	0	Undefined

For the reciprocal functions, inspect the denominator:

• At 0° and 180°, cosecant and cotangent are undefined because y = 0. Secant is 1 and -1 respectively.
• At 90° and 270°, secant is undefined because x = 0. Cosecant is 1 and -1 respectively, while cotangent is 0 at both angles.

Zero is neither positive nor negative. Undefined is not a number, so do not report it as zero or as infinity in an exact-value answer.

Negative angles and reflected points

Take a point (x, y) reached by an angle θ. Reflect it across the x-axis. The new point is (x, -y), corresponding to -θ.

The horizontal coordinate stays the same, while the vertical coordinate changes sign. This gives:

cos(-θ) = cos θ and sin(-θ) = -sin θ.

Also, tan(-θ) = -tan θ whenever tangent is defined. Secant follows cosine; cosecant and cotangent follow the sign reversal, wherever those functions exist.

For example, cos(-60°) = 1/2 while sin(-60°) = -sqrt(3)/2. The angle is negative in both expressions, but the two answers have different signs. The reflected point explains why.

Mistakes that a coordinate check catches

Using a reference angle’s sign for the original angle. The reference angle is acute, but the original angle might end in any quadrant. For cos 120°, the reference angle is 60°; the answer is -1/2 because the endpoint is on the left.

Assuming one ratio determines the quadrant. A positive sine leaves I and II possible. You need another sign or an interval to decide between them.

Taking a positive square root automatically. If cos²θ = 9/25, then cos θ can be 3/5 or -3/5. Use the supplied interval to choose. If no information settles the sign, keep both possibilities.

Choosing the sign of a changed angle from the original angle. If θ is in quadrant II, θ/2 is in quadrant I. But 2θ ranges from 180° to 360°, so it can lie in III, on the negative y-axis, or in IV. Find the target angle’s interval first. This is especially helpful when working with double, triple, and half-angle formulae.

Mixing degree and radian boundaries. The number 4 without a degree symbol can represent 4 radians. Since π < 4 < 3π/2, that angle is in quadrant III. It is very different from 4°, which lies in I.

Try these before reading the answers

For sign questions, a positive or negative answer is enough. For exact-value questions, keep fractions and square roots.

1. Which functions are positive at 318°?
2. Find the sign of cos(-205°).
3. Evaluate sin 510°.
4. Evaluate cot(7π/4).
5. If cos θ = -7/25 and θ is in quadrant III, find sin θ and tan θ.
6. If sec θ > 0 and cot θ < 0, identify the quadrant.
7. Find sec 180°, cosec 270°, and cot 90°.
8. Can sin θ > 0, cos θ < 0, and tan θ > 0 hold for the same real angle?
9. Find the sign of sin 4, with 4 measured in radians.
10. If 90° < θ < 180°, determine the sign of cos(θ/2).

Answers with quick reasons

1. Cosine and secant. 318° lies in IV, where x is positive and y is negative.
2. Negative. -205° + 360° = 155°, which lies in II.
3. 1/2. 510° reduces to 150°; sine is positive with reference angle 30°.
4. -1. 7π/4 lies in IV; its sine and cosine have equal magnitudes and opposite signs.
5. sin θ = -24/25; tan θ = 24/7. Both coordinates are negative in III, and 25² - 7² = 24².
6. IV. Positive secant requires positive cosine; negative cotangent then requires negative sine.
7. -1, -1, and 0 respectively. Use the corresponding axis coordinates.
8. No. Positive sine divided by negative cosine makes tangent negative.
9. Negative. π < 4 < 3π/2, so the endpoint lies below the x-axis in III.
10. Positive. Dividing the interval by 2 gives 45° < θ/2 < 90°, inside I.

If an answer went wrong, identify which decision failed: reducing the angle, locating the endpoint, choosing the function’s coordinate, or calculating the magnitude. Repairing that one step is more useful than copying the whole chart again.

Frequently asked questions

What is the quickest way to remember trigonometric signs?

Remember cos θ = x and sin θ = y on the unit circle. Right means positive cosine; above means positive sine. Tangent is positive when x and y have the same sign. Reciprocal functions follow their corresponding signs where defined.

Which trigonometric functions are positive in each quadrant?

All six are positive in I. Sine and cosecant are positive in II. Tangent and cotangent are positive in III. Cosine and secant are positive in IV. The other functions are negative within each open quadrant.

Why is tangent positive in the third quadrant?

Both sine and cosine are negative there. Their quotient, sin θ / cos θ, is positive because a negative number divided by a negative number is positive.

Does a negative angle always give a negative answer?

No. A negative angle specifies clockwise rotation. The terminal point determines the function’s sign. For example, sec(-300°) = 2.

Are angles of 90° and 180° part of a quadrant?

They lie on quadrant boundaries, not inside quadrants. Use the unit-circle coordinates to calculate their values, including any zero or undefined results.

Why are sine and cosecant always the same sign when both exist?

Cosecant is the reciprocal of sine. Taking the reciprocal of a nonzero real number preserves its sign. When sine is zero, cosecant is undefined.

Can I find a quadrant from tangent alone?

Its sign narrows the options to two quadrants. Positive tangent means I or III; negative tangent means II or IV. Another condition is needed to choose one.

Is the reference angle measured from the y-axis?

For this method, it is measured from the x-axis. For example, 110° has reference angle 70°, not 20°. The complementary 20° can be useful with other identities, but it is not the reference angle used here.

How do I handle an angle larger than 360° or 2π?

Remove complete turns until you have an angle within one turn. The original angle and the reduced angle share their terminal side and trigonometric values wherever defined.

Why does a square-root calculation sometimes need a negative sign?

Squaring removes sign information. If sin²θ = 16/25, sine may be 4/5 or -4/5. A quadrant or interval condition tells you which choice fits the original angle.