Sine Calculator

What sine measures

Sine belongs to right triangle trigonometry. Pick one of the two acute angles, and the sine of that angle is the length of the side opposite it divided by the length of the hypotenuse. Because both lengths scale together, the ratio depends on the angle and nothing else: a 30 degree triangle is a 30 degree triangle whether the hypotenuse is 6 centimeters or 6 kilometers.

The unit circle gives the same ratio a second meaning. Slide an angle counterclockwise around a circle of radius 1, and the sine of the angle is the height of the point you land on. That picture explains the full range from minus 1 to 1 and why the values repeat every full turn.

Those five values cover a surprising share of homework and shop-floor problems, and the curve they sketch is the first quarter of the familiar sine wave.

How to use the calculator

Type an angle into the box, choose degrees or radians from the selector, and the calculator returns the sine to four decimal places, with trailing zeros trimmed. It also shows the cosecant, which is the reciprocal: one divided by the sine. At 30 degrees the sine is 0.5 and the cosecant is 2; at 90 degrees both collapse to 1.

If you clear the box, the calculator shows the message Enter an angle to see its sine, so an empty state never produces a misleading zero. A sine of exactly 0 is displayed as 0, with the cosecant marked undefined, because dividing by zero has no answer.

Entry formats it accepts: plain decimals like 17.5, negative angles like -30, and any number of degrees past 360, which it evaluates on the wrapped angle. Radians work the same way, including values greater than 2 times pi.

The degree-versus-radian switch is the single biggest source of wrong answers on this kind of tool, so the next section spends a moment on it.

Degrees, radians and the input switch

The same number means different angles depending on the unit. In degrees, a full turn is 360. In radians, a full turn is 2 times pi, and a half turn is pi, which is why 180 degrees equals pi radians. One radian works out to about 57.3 degrees, and one degree to about 0.01745 radians.

The common conversions are worth memorizing: 30 degrees is one sixth of pi, about 0.5236 radians; 45 degrees is one quarter of pi, about 0.7854 radians; 90 degrees is pi over 2, about 1.5708 radians.

Why radians exist at all: in radian measure, arc length equals radius times angle, which makes calculus formulas cleaner and physics equations shorter. Engineering and programming libraries default to radians for that reason.

Practical advice: if a result looks wildly wrong, check the selector before rechecking the math. The calculator on this page evaluates sine of the raw number you type under the unit you selected, so 30 under radians and 30 under degrees give different answers by design.

Worked examples: ladders, ramps and roof pitch

Sine turns an angle and a length into a height. A 6 meter ladder leans against a wall at 30 degrees from the ground. The wall is the opposite side and the ladder is the hypotenuse, so the ladder's base sits 6 x 0.5 = 3 meters up the wall.

A loading ramp rises at 20 degrees and runs 10 feet along the slope. Its top edge stands 10 x 0.342 = 3.42 feet above the pavement, since the sine of 20 degrees rounds to 0.342. Dock workers use the same figure when they check that a ramp stays under a safe grade for forklifts.

The reciprocal direction works too. If you know the height and need the angle, that is the inverse sine: rise over run's hypotenuse first, then take arcsin of the ratio. A ramp that climbs 1 meter over a 12 meter slope length has a sine of 1 / 12 = 0.0833, and the arcsine of that lands near 4.8 degrees.

Notice the pattern in all three: sine connects the angle to a ratio of two sides. Measure any two of angle, height and slope length, and the third follows.

The unit circle, negative angles and full turns

On the unit circle, sine is the vertical coordinate of the point at your angle. That single picture explains three behaviors that trip people up on tests.

First, negative angles. Clockwise rotation puts the point below the horizontal axis, so the sine flips sign: the sine of minus 30 degrees is minus 0.5, the mirror of the positive case.

Second, angles past 90 degrees. Between 90 and 180 degrees sine stays positive but falls from 1 back to 0. Past 180 degrees the point dips below the axis and sine turns negative, reaching minus 1 at 270 degrees before climbing home.

Third, repetition. A full turn returns the point to where it started, so sine repeats every 360 degrees: the sine of 390 degrees matches the sine of 30 degrees, 0.5. The same holds in radians with a period of 2 times pi.

For a guided visual version, the trigonometry lessons at Khan Academy build the unit circle from scratch, and the free OpenStax Algebra and Trigonometry book covers the same ground with exercises.

Sine waves in the real world

Spinning machinery generates sine waves naturally. Household alternating voltage follows a sine in time: it rises to a peak, falls through zero, dips to a negative peak and returns, 60 times per second on a 60 Hz grid. One full cycle therefore takes 1 / 60 = 0.0167 seconds, and on a 50 Hz system the cycle is 1 / 50 = 0.02 seconds.

AC voltages carry a sine-specific quirk: the number on the label is not the peak. A 120 V wall outlet reaches 120 x 1.4142 = 169.7 volts at its peak, because 120 is the root-mean-square value, which sits at 0.7071 of the peak, and that factor is the sine of 45 degrees. European outlets labeled 230 V peak near 325 volts by the same logic.

Sound works the same way: a pure tone is a sine wave in air pressure, its frequency sets the pitch, and its amplitude sets the loudness. Tuning forks and test-tone generators produce clean sines precisely because every other waveform is a sum of them.

Cosecant and the inverse sine

Two relatives of sine answer different questions, and the calculator shows one of them.

Cosecant, written csc, is the reciprocal: csc equals 1 divided by the sine. It answers how stretched the triangle is for a given angle. At 30 degrees, 1 / 0.5 = 2; at 45 degrees it is about 1.4142; at 90 degrees it equals 1. It grows without bound as the angle approaches 0, which is why cosecant has no value at 0 degrees and 180 degrees, where the sine hits zero.

Inverse sine, or arcsin, runs the machine backwards: give it a ratio between minus 1 and 1 and it returns the angle whose sine that is. The arcsin of 0.5 is 30 degrees. The output is restricted to minus 90 through 90 degrees because infinitely many angles share each sine value, and a function must return one answer.

Keep the two straight with this test: if the input is an angle, you are computing sine or cosecant; if the input is a ratio, you are computing an inverse. This site's inverse-sine calculator handles the second case, and the trigonometry calculator evaluates the whole family at once.

Common sine mistakes

Most wrong sine answers come from one of the slips below, and all of them are easy to catch.

The mode error deserves one more line because it is so quiet: the answer comes back as a plausible decimal, with no error message. Sine of 30 radians is about minus 0.988 while sine of 30 degrees is 0.5, and nothing about either number announces which mode produced it. Glance at the selector whenever the result surprises you.

Rounding too early is the second quiet failure. Carrying 0.7071 through a five-step calculation instead of the full value can shift the final digit, so keep extra places in intermediate work and round once at the end.