One of the sixteen angles with exact values. In the first quadrant, so the signs follow from the quadrant.
All six functions
Each trigonometric function at this angle, with its exact value where one exists and what it means
Function
Exact
Decimal
What it is
sin
1/2
0.5
the y-coordinate on the unit circle
cos
√3/2
0.8660254038
the x-coordinate on the unit circle
tan
√3/3
0.5773502692
sin ÷ cos
csc
—
2
1 ÷ sin
sec
—
1.154700538
1 ÷ cos
cot
—
1.732050808
cos ÷ sin
Quadrant
Q1
the first quadrant
Within one turn
30°
the coterminal angle
In radians
0.52359878
degrees × π/180
Exact values?
yes
one of the sixteen
30° is one of the sixteen angles with exact values. The surd column is the answer an exam wants; the decimal beside it is rounded.
The three reciprocal functions pair with the ones you would expect only if you read them carefully: cosecant is 1/SIN and secant is 1/COS, not the other way round. The "co" in cosecant does not point at cosine.
All six are periodic. sin and cos repeat every 360°, tan and cot every 180°, so this angle shares its values with 30° + 360n — and tan additionally with 210°.
An undefined value is reported as undefined rather than as 1.6 × 10¹⁶, which is what a double actually returns for tan 90°.
What this tool shows
sin 30° is 1/2 exactly, not 0.4999999999999999. Sixteen angles have values you can write down precisely, and this page gives those in surd form beside the decimals — along with the three reciprocal functions almost nobody remembers the definitions of.
Sine, cosine and tangent at any angle
Cosecant, secant and cotangent
Exact surd values at the sixteen special angles
Which quadrant the angle lands in
The coterminal angle within one turn
Where a function is undefined, and why
All six functions Exact surd values Degrees or radians Signs by quadrant
Exact surds where they exist; undefined reported as undefined.
Updated 7 September 2026 · Works in any browser, no installation
They are ratios read off the unit circle. cos θ is the x-coordinate of the point at angle θ, sin θ is the y-coordinate, and tan θ is their quotient. The other three are reciprocals: csc = 1/sin, sec = 1/cos, cot = 1/tan — and cosecant pairs with SINE despite the name.
At a glance
Formula shown
On the unit circle, cos \u03b8 is the x-coordinate and sin \u03b8 is the y-coordinate. tan \u03b8 = sin \u03b8 / cos \u03b8. The reciprocals are csc \u03b8 = 1/sin \u03b8, sec \u03b8 = 1/cos \u03b8 and cot \u03b8 = 1/tan \u03b8 \u2014 note that cosecant pairs with sine, not cosine.
Scenario support
Evaluating a trig function for a geometry or physics problem; checking a calculator answer against the exact value; finding which functions are positive in a given quadrant.
Educational estimate
Planning support from the values you enter — not professional advice.
What the six functions are
Three primary and three reciprocal, and the naming is genuinely unhelpful.
sin, cos, tan. On the unit circle, cosine is the x-coordinate, sine is the y-coordinate, and tangent is sine over cosine — the slope of the ray.
csc, sec, cot. The reciprocals. And here is the trap: cosecant is 1/sine, not 1/cosine. Secant is 1/cosine. The “co” in cosecant does not point at cosine, and the pairing is the opposite of what the names suggest.
The mnemonic that works: the reciprocal of a function starting with “co” does not, and vice versa. Cotangent is the exception that proves nothing — it is 1/tan, and both names look related because they are.
Why tan 90° is undefined
tan θ is sin θ divided by cos θ. At 90° the cosine is exactly zero, so the division has no answer.
“Infinity” is not the answer either. Approaching 90° from below the tangent grows without bound; approaching from above it falls without bound. The two sides disagree, so there is no single value to assign — which is precisely what undefined means.
Here is what makes this a numerical problem rather than only a conceptual one. A double cannot represent π/2 exactly. Compute tan(Math.PI/2) and you get about 1.633 × 10¹⁶ — a perfectly finite number, and completely wrong as an answer to “what is tan 90°”.
So this page decides on the ANGLE, before any division happens. If the angle is an odd multiple of 90°, tangent and secant are reported as undefined. If it is a multiple of 180°, cosecant and cotangent are. That is the only way to get it right.
The exact values worth knowing
Sixteen angles have values you can write exactly, and they all come from two triangles.
From the 30-60-90 triangle: sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, cos 60° = 1/2.
From the 45-45-90 triangle: sin 45° = cos 45° = √2/2.
On the axes: sin 0° = 0, cos 0° = 1, and the pattern rotates.
The other angles — 120°, 135°, 150° and the rest — are these with signs from the quadrant. That is sixteen values from two triangles and one sign rule, which is why memorising the table is unnecessary if you can draw the triangles.
Signs by quadrant
The size of a trigonometric value comes from the reference angle; the sign comes from the quadrant.
All Students Take Calculus, read anticlockwise: All positive in the first quadrant, Sine in the second, Tangent in the third, Cosine in the fourth.
It is not arbitrary. Cosine is the x-coordinate and sine the y-coordinate, so in the second quadrant x is negative and y positive — cosine negative, sine positive, and their ratio negative too. In the third both coordinates are negative, so the ratio is positive, which is why tangent survives there.
The reciprocals follow their partners exactly: cosecant has the sign of sine, secant of cosine, cotangent of tangent.
Degrees or radians
The same angle, two units. 180° is π radians, so a degree is π/180 radians and a radian is 180/π ≈ 57.29578°.
Degrees are older and more intuitive; the 360 comes from Babylonian astronomy and divides neatly by a great many numbers.
Radians are the ones mathematics actually uses, because they make the formulas simple. The derivative of sin x is cos x only in radians; in degrees it picks up a factor of π/180. Arc length is rθ only in radians. Every series expansion assumes them.
A practical rule: geometry and surveying in degrees, calculus and physics in radians, and always check which mode a calculator is in before trusting an answer.
Periodicity
All six functions repeat, and knowing the period tells you which other angles share a value.
sin, cos, csc, sec repeat every 360°. The ray comes back to where it started after a full turn.
tan and cot repeat every 180°. Half a turn puts the ray on the opposite side of the origin, which negates both coordinates and leaves their ratio unchanged.
That shorter period for tangent is exactly why arctan returns values over a 180° window while arcsin and arccos use 180° windows of their own — each inverse has to cover one full period of its function and no more.
Sources and methodology
The definitions and the exact values are curriculum standards; these are the references.
Method. Special angles are matched against a table of exact surd values rather than reverse-engineered from a decimal, so 1/2 is reported as 1/2. Whether a function is undefined is decided on the ANGLE rather than on the computed ratio: a double cannot represent π/2, so tan 90° evaluates to about 1.6 × 10¹⁶ and testing the output would report a very large number instead of no number at all. That engine is verified on every change against 71 hand-written assertions, including that sin²θ + cos²θ = 1 across ten thousand angles, that tan really is sine over cosine wherever it is defined, and that each reciprocal function multiplies its partner to exactly 1. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Inverse Trigonometric FunctionsThe principal value AND the general solution, because a calculator hands you one member of an infinite family and the context decides which one your problem wants.
Unit CircleWhere an angle lands, with exact surd coordinates and the radian measure as a multiple of π — because the coordinates ARE the cosine and sine.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
Double and Half Angle2θ and θ/2 on one page, with all three forms of cos 2θ evaluated and the half-angle ± resolved from the quadrant of θ/2 — not of θ.
TriangleSolve any triangle from three measurements — and when two sides and a non-included angle describe TWO triangles, this one shows both instead of picking one.
Special Right Triangles30-60-90 and 45-45-90 from any one side, in exact surd form as well as decimals — with the warning that the long leg is √3 times the short, not twice.
The Unit Circle Calculator draws where these values come from, and the Inverse Trigonometric Functions Calculator runs the other way — from a value back to an angle, with the general solution.
Educational use disclaimer
This is an educational tool. Exact values are exact; the decimals beside them are rounded, and an undefined value is reported as undefined rather than as a very large number.
Published the trig functions page giving all six at once with exact surd values at the sixteen special angles, since the surd is what an exam wants and a decimal hides the structure.
An undefined value is reported as undefined: tan 90° computes as about 1.6 × 10¹⁶ in double precision because π/2 is not representable, so the decision is made on the angle rather than on the ratio.
Names the trap in the reciprocal names — cosecant is 1/sin and secant is 1/cos, and the 'co' does not point where it looks like it does.
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