What are sin, cos and tan of 240 degrees?

Sine of 240° is −√3/2, or -0.866025 as a decimal. Cosine is −1/2, which is -0.500000. Tangent is √3, or 1.732051. The reference angle is 60°, so all three magnitudes are the 60° magnitudes with signs set by quadrant 3.

What are csc, sec and cot of 240 degrees?

Cosecant is 1 over sine, giving -1.154701. Secant is 1 over cosine, giving -2.000000. Cotangent is 1 over tangent, giving 0.577350. At least one of these lands on an exact whole number, which happens whenever the function it inverts is exactly one half.

What is 240 degrees in radians?

240° is 4π/3 radians, which is 4.188790 as a decimal, and 266.6667 gradians. The conversion is degrees times π divided by 180, since a full turn of 360° is 2π radians. Put another way, 240° is 66.6667% of a complete revolution.

Which quadrant is 240 degrees in?

Quadrant 3, with a reference angle of 60°. That means the acute angle between the radius and the horizontal axis is 60°, so every value here has the same magnitude as the corresponding value at 60°, differing only in sign. In quadrant 3 the positive functions are tangent and cotangent.

How does this connect to waves?

Directly. Plot the vertical coordinate of a point moving steadily round the unit circle against time and you have drawn a sine wave — the circle is the definition and the wave is its shadow. That is why the same functions describe alternating current, sound, light, tides and any oscillation at all. The angle becomes a phase, a full turn becomes one cycle, and the periodicity you see in these values becomes the frequency.

What is the fastest way to memorise all of this?

Learn the first quadrant and one triangle. The 30-60-90 triangle with sides 1, √3 and 2 gives every value at 30° and 60°; the 45-45-90 triangle with sides 1, 1 and √2 gives 45°. Then learn the axis values at 0° and 90° from the unit circle, which needs no triangle at all. Everything else is a reference angle and a sign, and no further memorisation is required or useful.

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.

Trigonometric Values of 240°

Two hundred and forty degrees is two thirds of a full turn, deep in the third quadrant, and it pairs with 60° exactly as 210° pairs with 30°. The reference angle is 60°, so the magnitudes are the familiar ones: √3/2, 1/2 and √3. Both coordinates are negative, which puts sine at −√3/2 and cosine at −...

ANGLE

UNIT

COMMON ANGLES

SIN 240°

-0.866025

EXACT

−√3/2

240.0000° = 4.188790 rad = 266.6667 grad · quadrant 3 · reference angle 60.00°

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HOW TO USE

1
This page opens with 240° already entered, so every value below is the answer for that angle — sine -0.866025, cosine -0.500000, and the rest — with no typing needed.
2
Read the exact form beside each decimal. At 240° sine is −√3/2 and cosine is −1/2, written as surds rather than rounded decimals.
3
Tap any of the six tiles to make it the headline figure. All six functions have finite values at 240°.
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 29, 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.