Each trigonometric function with its sign in this quadrant and the reason
Function
Sign
Why
sin
+
positive in the second quadrant
cos
−
negative in the second quadrant
tan
−
negative in the second quadrant
All Students Take Calculus — All in the first quadrant, Sine in the second, Tangent in the third, Cosine in the fourth.
The reference angle is the acute angle between the ray and the x-axis — never the y-axis, which is the usual slip.
Every trigonometric value of the original angle equals the value of its reference angle, up to a sign the quadrant decides. That is what makes reference angles worth computing at all.
Degrees are carried as exact fractions, so 247.5° and 1/3 of a degree both reduce without drift.
What this tool shows
The reference angle of 225° is 45°, because 225 − 180 = 45. It is always measured to the x-axis, always between 0° and 90°, and it is what lets you find any trigonometric value from the first quadrant plus a sign.
The reference angle of any angle in degrees
Which quadrant the angle lands in
The rule for that quadrant, with your numbers in
The coterminal angle within one turn
Whether sine, cosine and tangent are positive there
Angles that are negative or past a full turn
The quadrant identified The rule substituted Signs of sin, cos, tan Negatives and big angles
Updated 7 September 2026 · Works in any browser, no installation
Reduce the angle into one turn, then measure the acute angle from the ray to the x-axis. For 225° that is 225 − 180 = 45°. The answer is always between 0° and 90°, and it is always to the horizontal axis — measuring to the vertical one is the mistake this page exists to prevent.
At a glance
Formula shown
Reduce to \u03b8 in [0\u00b0, 360\u00b0). Then the reference angle is \u03b8 in the first quadrant, 180\u00b0 \u2212 \u03b8 in the second, \u03b8 \u2212 180\u00b0 in the third, and 360\u00b0 \u2212 \u03b8 in the fourth. It always lands between 0\u00b0 and 90\u00b0.
Scenario support
Evaluating sin 225\u00b0 from sin 45\u00b0; checking a unit-circle exercise; reducing an angle before looking up a trig value.
Educational estimate
Planning support from the values you enter — not professional advice.
What it actually is
Draw the angle from the positive x-axis. Wherever the ray ends up, the reference angle is the acute angle between that ray and the x-axis — the nearest bit of horizontal, whether that is to the left or the right.
It is always between 0° and 90°, because it is the acute angle. And it is always to the x-axis: measuring to the y-axis gives the complement, which is a different number and a different idea.
The point of it is that trigonometric values repeat. sin 225° and sin 45° differ only by a sign, because the two rays make the same angle with the horizontal. Knowing the first quadrant plus a sign is enough to know everything.
The four quadrant rules
First quadrant (0° to 90°): the reference angle is the angle itself. The ray is already close to the x-axis on the right.
Second (90° to 180°): 180° − θ. At 150° the ray is 30° above the negative x-axis, so the reference angle is 30°.
Third (180° to 270°): θ − 180°. At 225° the ray is 45° past the negative x-axis, so 45°.
Fourth (270° to 360°): 360° − θ. At 300° the ray is 60° short of the positive x-axis, so 60°.
Each rule is the same instruction: how far is this ray from the nearest horizontal? The four forms differ only because of which horizontal is nearest and which way round the subtraction goes.
Why it is worth computing
Without reference angles you would need a table covering every angle from 0° to 360°. With them you need one covering 0° to 90°, and a rule for signs.
sin 225° = −sin 45° = −0.7071. The size comes from the reference angle; the minus sign comes from being in the third quadrant, where sine is negative.
This is exactly how trig tables worked before calculators, and it is still how the unit circle is taught, because it turns an infinite set of angles into ninety degrees plus bookkeeping.
It is also why the exact values worth memorising — 30°, 45°, 60° — are enough. Every angle whose reference angle is one of those has an exact trigonometric value too.
ASTC, and what it means
All Students Take Calculus, read anticlockwise from the first quadrant: All three positive in the first, Sine in the second, Tangent in the third, Cosine in the fourth.
It is not arbitrary. On the unit circle, cosine is the x-coordinate and sine is the y-coordinate. In the second quadrant x is negative and y is positive, so cosine is negative and sine is positive. Tangent is their ratio, so it is negative there too.
In the third quadrant both coordinates are negative, so sine and cosine are both negative and their ratio is positive — which is why tangent is the survivor there. The mnemonic is a shortcut for reading off two signs.
The page reports the signs directly for whichever quadrant your angle lands in, so the mnemonic is a check rather than a dependency.
Negatives and angles past 360°
A negative angle just means turning clockwise. −45° ends up in the same place as 315°, so its reference angle is 45°.
An angle past 360° has gone round more than once. 750° is two full turns plus 30°, so it has the same reference angle as 30°.
Both cases reduce the same way: find the coterminal angle between 0° and 360° first, then apply the quadrant rule. This page does the reduction by exact division rather than by adding 360 repeatedly, so a million-degree angle costs nothing and stays exact.
The mistakes worth naming
Measuring to the y-axis. The commonest one. The reference angle of 150° is 30°, not 60°. It is always the horizontal.
Forgetting to reduce first. Applying a quadrant rule to 430° directly gives nonsense. Reduce to 70° and then apply.
Carrying the sign into the reference angle. The reference angle is never negative. It is a size. The sign belongs to the trigonometric value, decided separately by the quadrant.
Assuming there is always a quadrant. 90°, 180° and 270° sit ON an axis, in no quadrant at all. Their reference angles are 90°, 0° and 90°, and one of the trig functions is undefined at each of them.
Sources and methodology
The convention is standard across curricula; these are the references.
Method. Degrees are carried as exact rationals, so the reduction to one turn is a single exact division rather than repeated addition, and 247.5° or a third of a degree behave the same as a whole number. The quadrant rule is rendered with the entered value substituted rather than shown as a static table. The suite compares every whole degree from −1080° to 1080° against an independently written routine that reduces by looping and applies the rules from a lookup. That engine is verified on every change against 82 hand-written assertions, including agreement with a loop-and-lookup routine on all 2,161 whole degrees between −1080° and 1080°, and that the reference angle never leaves [0°, 90°]. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Coterminal AngleThe principal angle between 0° and 360°, the nearest negative one, and the family θ + 360n they all belong to — reduced by exact division, not by looping.
Complementary and Supplementary AnglesBoth at once, plus the explement — and an honest answer when there is none, since an obtuse angle has no complement and a negative number is not one.
Angle Between Two VectorsDot product, magnitudes and the angle as separate steps, in the plane or in space — with exactly parallel vectors returning 0° rather than a floating-point smudge.
Central AngleLeave one of radius, arc and angle blank and the page solves for it — with the sector drawn, plus chord, sector area and segment area from the same angle.
Hyperbolic FunctionsAll six hyperbolic functions and their inverses, built visibly from e^x and e^-x, with cosh squared minus sinh squared computed rather than asserted and every domain limit stated.
Clock AngleThe angle between the hands with the hour hand where it really is — at 3:30 that is 75°, not the 90° most people answer — and the face drawn to show which angle it is.
If you only need the angle reduced into one turn and not the acute angle to the axis, the Coterminal Angle Calculator does that step and gives the whole family of equivalent angles.
Educational use disclaimer
This is an educational tool. Degrees are carried as exact fractions, so 247.5° reduces without drift; the trigonometric values themselves are not computed here, only their signs.
Published the reference angle page printing the quadrant rule with the reader's own numbers substituted, since a static table is what lets the y-axis mistake survive.
The sign of sine, cosine and tangent in that quadrant is given alongside, because a sign is the only reason to want a reference angle in the first place.
Checked against an independently written loop-and-lookup routine on all 2,161 whole degrees from −1080° to 1080°.
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