What are sin, cos and tan of 0 degrees?

Sine of 0° is 0, or 0.000000 as a decimal. Cosine is 1, which is 1.000000. Tangent is 0, or 0.000000. This angle sits on an axis rather than inside a quadrant, so its values are the extremes: exact integers and zeros rather than surds.

Why are csc and cot undefined at 0 degrees?

Because they are divisions and the denominator is exactly zero. Cosecant is 1 over sine, and sine of 0° is exactly 0. Cotangent is cosine over sine, so it divides by the same zero. Neither has a value — not a very large one, but none at all. The remaining functions are perfectly well behaved: sin 0° = 0.000000, cos 0° = 1.000000, tan 0° = 0.000000, sec 0° = 1.000000.

What is 0 degrees in radians?

0° is 0 radians, which is 0.000000 as a decimal, and 0.0000 gradians. The conversion is degrees times π divided by 180, since a full turn of 360° is 2π radians. Zero is the one angle that is the same number in every unit.

Why does 0 degrees not belong to a quadrant?

Because it lies exactly on an axis rather than between two of them. The quadrants are the four open regions between the axes, and 0° points straight along one, so it belongs to none. These boundary angles are where the extreme values live — a sine or cosine of exactly 0, 1 or −1 — and where the functions that divide by them stop existing. Its reference angle is 0°.

Why is one value exact and another a long decimal?

Because some trigonometric values are rational or simple surds and others are not. Sine of 30° is exactly one half, a number that can be written down completely. Sine of 40° is an irrational number with no closed form in surds, so any decimal is a truncation. Every value at a multiple of 30° or 45° falls in the first category, which is precisely why those angles are the ones worth memorising — you can write them exactly rather than approximately.

What does the unit circle actually show?

A circle of radius exactly 1, centred on the origin. Sweep an angle anticlockwise from the positive horizontal axis and the point where the radius meets the circle has coordinates (cos θ, sin θ). That is the definition, not an illustration of it. Because the radius is 1 the hypotenuse of the little right triangle is 1, so the adjacent side simply is the cosine and the opposite side simply is the sine — no division needed.

How do the signs change from quadrant to quadrant?

They follow the coordinates. In the first quadrant both coordinates are positive so everything is positive. In the second the horizontal goes negative, so cosine, secant and tangent turn negative. In the third both are negative, so sine and cosine are negative but tangent — their ratio — comes back positive. In the fourth only the vertical is negative, so sine, cosecant and tangent are negative. The usual mnemonic, All, Sine, Tangent, Cosine, names the function that stays positive in each quadrant in turn.

What is a reference angle for?

It reduces every angle to a first-quadrant one. The reference angle is the acute angle between the terminal radius and the horizontal axis, always between 0° and 90°. Every trigonometric value of any angle equals the value of its reference angle with a sign supplied by the quadrant. Learn the first quadrant properly and the remaining three become sign bookkeeping rather than new material.

Why do some functions have no value at certain angles?

Because they are defined as divisions, and the denominator sometimes hits zero. Tangent and secant both divide by cosine, so both are undefined wherever cosine is zero — at 90° and 270°. Cosecant and cotangent divide by sine, so both are undefined where sine is zero — at 0°, 180° and 360°. Undefined does not mean very large. It means there is no such number, and a calculator reporting one is reporting its own rounding error.

Trigonometric Values of 0°

Zero degrees is where the whole subject starts, and it is the one angle whose values you can read straight off the picture without computing anything. The point sits at the far right of the unit circle, at coordinates (1, 0), so cosine — the horizontal coordinate — is exactly 1, and sine — the verti...

ANGLE

UNIT

COMMON ANGLES

SIN 0°

0.000000

EXACT

0.0000° = 0.000000 rad = 0.0000 grad · on an axis · reference angle 0.00°

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HOW TO USE

1
This page opens with 0° already entered, so every value below is the answer for that angle — sine 0.000000, cosine 1.000000, and the rest — with no typing needed.
2
Read the exact form beside each decimal. At 0° the values are whole numbers and zeros rather than surds, because the angle lies on an axis.
3
Tap any of the six tiles to make it the headline figure. Note that csc and cot are marked undefined here — that is the correct answer, not a failure to compute.
4
Change the angle in the box, or press play, to turn this into the full trigonometry engine — right triangles and any triangle from SSS, SAS, ASA, AAS or SSA are all in the same tool.

THE FORMULAS

Sinesin A = opposite ÷ hypotenuse

Cosinecos A = adjacent ÷ hypotenuse

Tangenttan A = opposite ÷ adjacent

Reciprocalscsc = 1/sin · sec = 1/cos · cot = 1/tan

Pythagorean identitysin²θ + cos²θ = 1

Law of sinesa/sin A = b/sin B = c/sin C

Law of cosinesa² = b² + c² − 2bc·cos A

Area from two sidesarea = ½·a·b·sin C

Heron's formulaarea = √(s(s−a)(s−b)(s−c))

CircumradiusR = a ÷ (2·sin A)

Inradiusr = area ÷ s

Angle sumA + B + C = 180°

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Last updated: July 28, 2026 · Formula verified · Every multiple of 15° is returned exactly, so tan 90° reads undefined rather than 1.6×10¹⁶ · Eagle-eyed accuracy for every calculation.