Cosecant Calculator

What the cosecant calculator returns

Cosecant answers one question: how much hypotenuse comes with one unit of opposite side. Divide 1 by the sine and the number is there: csc(theta) = 1 / sin(theta). At 30 degrees the sine is 0.5, so 1 / 0.5 = 2, and the answer carries no units because both lengths cancel.

Three input modes cover the ways people arrive at this function. Type an angle in degrees or radians, and a form like pi/6 also works, to see the cosecant with the sine it came from. Type a sine value to get the cosecant plus both angles between 0 and 360 degrees that share that sine. Enter the opposite side and the hypotenuse of a right triangle to get the ratio straight from the two measurements, which is the form riggers and carpenters use.

One rule holds in every mode: the size of a cosecant is never below 1. Sine lives between -1 and 1, and the reciprocal of a number in that band lands at or outside it. Every real answer is at least 1 or at most -1, and the sign always matches the sign of the sine.

Section 4.14 of the NIST Digital Library of Mathematical Functions defines csc as 1 over sin and lists the zeros of sine in the same place, which is why the undefined angles on this page line up with the zeros of the sine curve.

Cosecant as a triangle ratio: rope per foot of height

In a right triangle the sine is opposite over hypotenuse, so the cosecant flips it to hypotenuse over opposite. Read the result as rope per foot of height. A guy wire from the top of a 30 ft mast to an anchor 40 ft from the base is 50 ft long, and 50 / 30 = 1.6667, so the cosecant of the angle at the anchor is 1.6667.

The angle itself is arcsin(30 / 50) = 36.87 degrees, and 1 / sin(36.87) = 1.6667 as well. Both routes agree because the side ratio and the angle carry the same information, and the calculator accepts either one.

The ratio tracks how much the wire leans. A wire that runs almost straight up has an opposite side close to the hypotenuse, so its cosecant sits just above 1, and a perfectly vertical wire would give csc 90 = 1. Move the anchor farther out and the hypotenuse grows while the height stays fixed, so the cosecant climbs. A wire twice as long as the mast is tall gives csc = 2, which is the same number as csc 30.

Exact values at the angles on the unit circle

The standard angles have exact cosecants, and every row can be checked by dividing 1 by the sine beside it. The exact column is worth knowing because it survives rounding: csc 45 is the square root of 2, not 1.414 alone, and csc 60 is 2 divided by the square root of 3, which is the same as 2 root 3 over 3.

Two patterns fall out of the table. Supplementary angles share a sine, so they share a cosecant: csc 30 and csc 150 are both 2, and csc 45 and csc 135 are both 1.4142. Angles in the third and fourth quadrants flip the sign, so csc 210 is -2 rather than 2. Off the standard list the arithmetic is the same: csc 40 is 1 / 0.6428 = 1.5557, and sin 56.31 is 0.8321, so its cosecant is 1.2019.

Where sine is zero: 0, 180, 360 and the asymptotes

Sine is zero at 0 degrees, at 180 degrees, at 360 degrees, and at every whole multiple of 180 in both directions. Dividing 1 by zero has no answer, so the cosecant has no value at those angles either. A graphing calculator draws a vertical dashed line there instead of a point, and that line is an asymptote.

The approach from each side is where the sign gets interesting. Just above 0 degrees the sine is small and positive, so the cosecant is huge and positive. Just below 0 degrees, which is 359 degrees, the sine is small and negative, so the cosecant is huge and negative. The two sides of the asymptote therefore run off in opposite directions, not the same one.

This is why a spreadsheet cell can show 5729 for an angle that was meant to be zero, and why the calculator on this page prints undefined when the input lands exactly on a pole. It is also why the curve repeats every 360 degrees and why csc(-40) = -csc(40): the function is odd, just like sine.

Cosecant, secant and cotangent: three reciprocals, three jobs

The three reciprocal functions are easy to mix up because they look alike and all blow up somewhere. Each one is tied to a different side of the same triangle, and that is the fastest way to keep them apart. Cosecant pairs with sine and therefore with the opposite side. Secant pairs with cosine and the adjacent side. Cotangent pairs with tangent and compares the two legs to each other.

Pick the function by the measurement you already have. If you know a height and need the length of a brace, you want cosecant, because it multiplies the height by the wire-to-height ratio. If you know a horizontal run and need the sloping length, you want secant. If you know both legs and need an angle for a miter or a grade, cotangent or its inverse does the job, and the slope calculator works the same ratio as rise over run.

The OpenStax Algebra and Trigonometry chapter on the other trigonometric functions graphs all three side by side and marks the same asymptotes, which is a useful check when a chart looks wrong.

Cosecant in the law of sines

The law of sines says a / sin A = b / sin B = 2R, where R is the radius of the circle that passes through all three corners of the triangle. Rewrite any term with a cosecant and the division turns into a multiplication: a = 2R sin A, and R = (a / 2) csc A. Before calculators, that second form was the reason cosecant tables existed at all, because multiplying by a tabulated value beat dividing by a decimal.

Worked example. A triangular lot has a side of 12 ft opposite a 30 degree corner. Then 2R = 12 / 0.5 = 24 ft, so the circumradius is 12 ft and the circle through the three corners is 24 ft across. If another corner measures 90 degrees, the side opposite it is 24 x sin 90 = 24 ft, which is the diameter, exactly as a right triangle requires.

The form also explains a common survey shortcut. Once 2R is known, every other side is 2R times the sine of its opposite angle, and every angle comes back from the side by arcsin. Cosecant is the multiplier that gets you from a known side and angle to the radius, and the radius is what makes the rest of the triangle fall out.

Three worked examples with real measurements

A tower from a transit. A transit sits 100 ft from the base of a 150 ft tower. The straight line from the instrument to the top is the square root of 100 squared plus 150 squared, which is 180.28 ft. The angle at the instrument is arctan(150 / 100) = 56.31 degrees, and the cosecant is 180.28 / 150 = 1.2019. The same angle has a sine of 0.8321, which is the reciprocal of that cosecant once both are carried to full precision.

A sine value without an angle. A sensor reports that a rope makes a sine of 0.25 with the vertical. Then csc = 1 / 0.25 = 4. Two angles between 0 and 360 degrees produce that sine: 14.48 degrees and 165.52 degrees, since 180 - 14.48 = 165.52. The calculator shows both, because a sine alone cannot tell you which side of vertical the rope is on.

A radian input. A physics problem gives the angle as 7pi/6. That is 210 degrees, its sine is -0.5, and its cosecant is -2. The same angle written as 3.6652 radians gives the same result, which is the check that the mode switch did what you expected.

Mistakes: radians, arcsin and the cotangent mix-up

The first mistake is the unit switch. A calculator left in radian mode reads 30 as 30 radians, which is 1718.87 degrees, or 278.87 degrees after four full turns, and the sine there is -0.98803. The cosecant comes out as -1.0121 instead of 2. If an answer is close to 1 when you expected a large number, check the mode before checking the formula.

The second is the notation. On almost every calculator, sin with a -1 superscript means arcsin, not 1 over sin. So sin inverse of 0.5 is 30 degrees, while the reciprocal of the number 0.5 is 2, and the cosecant of 0.5 radians is 2.0858. Three different answers, one keystroke apart, and the calculator keyboard offers no hint about which one you pressed.

The third is the cotangent. Cotangent is cosine over sine, which is 1 over tangent, and it is not the reciprocal of cosecant. At 45 degrees csc is 1.4142 and cot is 1.0000; at 30 degrees csc is 2 and cot is 1.7321. Cotangent shares the asymptotes at 0, 180 and 360 with cosecant, which is where the confusion starts, but its values are different everywhere else.

The fourth is the sign. A negative sine gives a negative cosecant, so csc 210 is -2 and not 2. Rounding causes the last error: 1 / sin(0.01 degrees) is 5729.578, and rounding the sine before dividing moves the answer by hundreds. Divide first, round last, and expect the number to change fast near the poles.