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MathSine Calculator
Compute sin(x) and arcsin(x) in degrees or radians. Unit-circle explanation, common-angle table, Law of Sines, and step-by-step examples for students and engineers.
Sine calculation
Inverse sine calculator
Have feedback? Report bugs, suggest features, or share your thoughts — we read them allThe Sine function ( sin(x) )
The sine function, written sin(x), is one of the three primary trigonometric functions. In a right triangle, sin(θ) equals the length of the side opposite angle θ divided by the length of the hypotenuse — the famous SOH in SOH-CAH-TOA. Beyond triangles, sine is the y-coordinate of a point that has been rotated by angle x around the origin on a unit circle, which is why the function repeats every full rotation. Sine appears everywhere: AC electricity, sound waves, light, ocean tides, pendulums, harmonic oscillators in quantum mechanics, and Fourier analysis of signals.
The output of sin(x) always lies between −1 and +1, and the graph traces a smooth wave. The key properties to remember are:
- Periodicity: sin(x) repeats every 2π radians (360°), so sin(x) = sin(x + 2kπ) for any integer k. This makes it ideal for modeling cyclic phenomena.
- Odd symmetry: sin(−x) = −sin(x). The curve is rotationally symmetric about the origin, unlike cosine which mirrors across the y-axis.
- Bounded range: −1 ≤ sin(x) ≤ 1 for every real x. This is why sine is used to model waves whose amplitude does not blow up.
On a unit circle (radius 1, centered at the origin), if you rotate the point (1, 0) by angle x counter-clockwise, the y-coordinate of the new point is sin(x) and the x-coordinate is cos(x). This geometric definition extends sine to every real number — positive, negative, or absurdly large — and is the foundation of how computers and calculators evaluate the function.
What is Degrees (deg °) and Radians (rad)?
Trigonometric functions accept angles in two standard units: degrees and radians. Mixing them up is one of the most common sources of wrong answers in physics and engineering homework, so it pays to understand the difference.
- Degrees: a full circle is split into 360 parts. The number 360 is historical — it has divisors 2, 3, 4, 5, 6, 8, 9, 10, 12, … which made fractions easy for Babylonian astronomers around 2000 BCE.
- Radians: a full circle is 2π ≈ 6.283 radians. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius. Radians are the natural unit in calculus because d/dx sin(x) = cos(x) only when x is in radians.
To convert between the two units, use these two formulas — they are inverses of each other:
- Degrees to radians: radians = degrees × π180
- Radians to degrees: degrees = radians × 180π
Table of common sine values
| Angle (°) | Angle (Radians) | sin(angle) | sin(angle) |
|---|---|---|---|
| 0° | 0 | 0 | 0.00 |
| 30° | π/6 | 1/2 | 0.50 |
| 45° | π/4 | √2/2 | 0.7071 |
| 60° | π/3 | √3/2 | 0.8660 |
| 90° | π/2 | 1 | 1.00 |
| 120° | 2π/3 | √3/2 | 0.8660 |
| 135° | 3π/4 | √2/2 | 0.7071 |
| 150° | 5π/6 | 1/2 | 0.50 |
| 180° | π | 0 | 0.00 |
| 210° | 7π/6 | -1/2 | -0.50 |
| 225° | 5π/4 | -√2/2 | -0.7071 |
| 240° | 4π/3 | -√3/2 | -0.8660 |
| 270° | 3π/2 | -1 | -1.00 |
| 300° | 5π/3 | -√3/2 | -0.8660 |
| 315° | 7π/4 | -√2/2 | -0.7071 |
| 330° | 11π/6 | -1/2 | -0.50 |
| 360° | 2π | 0 | 0.00 |
Frequently Asked Questions
Draw an equilateral triangle with all sides of length 2 and all angles of 60°. Cut it in half straight down through one vertex: you get two identical right triangles, each with a 30° angle at the top, a 60° angle at the bottom-left, and a 90° angle where the cut meets the base. The hypotenuse is still 2 (the original side), and the side opposite the 30° angle is exactly half the base, which is 1. By the SOH definition, sin(30°) = opposite ÷ hypotenuse = 1 ÷ 2 = 1/2. This same triangle gives cos(30°) = √3/2 (using Pythagoras on the remaining side) and tan(30°) = 1/√3. These are not approximations — they are exact rational and irrational values that students are expected to memorize, and they show up constantly in physics problems involving inclined planes and projectile motion at 30° or 60°.
sin(x) takes an angle and returns a ratio between −1 and 1. sin⁻¹(x), also written arcsin(x) or asin(x), takes a ratio between −1 and 1 and returns an angle. They are inverse operations, so sin(arcsin(0.5)) = 0.5 and arcsin(sin(30°)) = 30°. The subtlety is that sin is not one-to-one — sin(30°), sin(150°), sin(390°) all equal 0.5 — so arcsin has to pick one canonical answer. By convention it returns the angle in the range [−90°, +90°] (or [−π/2, +π/2] in radians). That is why this calculator displays two arcsin results: the principal value and the supplementary angle 180° − principal, both of which have the same sine. Notation warning: sin⁻¹(x) does NOT mean 1/sin(x); the reciprocal is csc(x), the cosecant. Confusing these two is a common test-killing mistake.
Mathematically sin(π) = 0 exactly, but π itself cannot be stored exactly in a computer — it is an irrational number with infinitely many digits, and IEEE-754 double-precision floating-point only keeps the first 15–17 significant digits. So when you type π, the calculator actually uses 3.141592653589793, which is slightly less than the real π. Sine of a number slightly less than π is a very small positive number (the Taylor series tells us it is approximately π − 3.141592653589793 ≈ 1.22e−16). The takeaway: any result with an exponent like e−15 or e−16 should be treated as zero, just rounded by the limits of binary arithmetic. Real numerical libraries handle this with high-precision argument reduction (the Payne–Hanek algorithm), but the underlying issue is fundamental: most real numbers are not representable in binary.
Use degrees when you are doing geometry, surveying, navigation, machining, carpentry, or anything where humans talk to humans about angles — "turn the wheel 45 degrees" is intuitive. Use radians when you are doing calculus, physics, or programming. The derivative rule d/dx sin(x) = cos(x) only works in radians; in degrees the derivative is (π/180)·cos(x), which is ugly. Likewise, the Taylor series sin(x) = x − x³/6 + x⁵/120 − … converges to the right answer only when x is in radians. Most programming languages (Python's math.sin, JavaScript's Math.sin, C's sin) expect radians by default, which is why graphing sin(45) in Python gives 0.851 instead of 0.707 — you forgot to convert. When in doubt, convert with degrees × π/180.
The Law of Sines says that in any triangle (not just right triangles), the ratio of each side to the sine of its opposite angle is the same: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the radius of the circumscribed circle. Use it when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA, the ambiguous case). It is the primary tool for solving non-right triangles in surveying, where you might measure two angles from a baseline to a distant tree and want the tree's distance. The SSA case is called "ambiguous" because a given side, side, and angle can sometimes describe two different triangles — for instance, two sides of 5 and 7 with an opposite angle of 30° can close into either a 38° or 142° triangle. Always sanity-check with the Law of Cosines when you suspect the ambiguous case bit you.
The right-triangle definition of sine only works for angles between 0° and 90° — you cannot have a triangle with a 200° angle inside it. The unit circle definition fixes this. Draw a circle of radius 1 centered at the origin. For any angle θ (positive going counter-clockwise from the positive x-axis, negative clockwise), the point on the circle at that angle has coordinates (cos θ, sin θ). For θ = 270°, the point is (0, −1), so sin(270°) = −1. For θ = −90°, the point is also (0, −1), so sin(−90°) = −1. For θ = 720° (two full loops), you are back at (1, 0), so sin(720°) = 0. The unit circle is what makes sine periodic, odd, and defined for all real numbers — properties that would be impossible to express with triangles alone.
Almost anywhere something oscillates. AC household electricity is a sine wave at 50 Hz (Europe) or 60 Hz (US): voltage(t) = 230 · sin(2π · 50 · t) volts. Pure musical tones are sine waves — a 440 Hz A above middle C is air pressure varying as sin(2π · 440 · t). Ocean tides are roughly the sum of a 12.42-hour sine (lunar) and a 12-hour sine (solar). A pendulum's angle follows sin for small swings. Light, radio, and Wi-Fi are all electromagnetic waves modeled by sines and cosines. The Fourier transform — used in JPEG, MP3, MRI scanners, voice recognition — decomposes any signal into a sum of sines and cosines of different frequencies. When engineers say "signal processing", they essentially mean "manipulating sine waves".
Calculators do not store a giant table of values; they use a two-step trick. First comes argument reduction: take the input x and subtract the largest multiple of 2π that keeps it in [0, 2π], then use symmetries (sin(π − x) = sin(x), sin(−x) = −sin(x), sin(π/2 − x) = cos(x)) to fold the angle into [0, π/4]. Second comes evaluation: in that small range, the Taylor series sin(x) = x − x³/6 + x⁵/120 − x⁷/5040 + … converges blazingly fast — six or seven terms give 15-digit accuracy. Pocket calculators historically used CORDIC (1959, Volder), which only needs addition, bit shifts, and a pre-computed table of arctan values — perfect for hardware that lacked a multiplier. Modern CPUs have fast multipliers, so they prefer polynomial approximations like the minimax remez polynomial. Either way, the answer you see is good to roughly half a unit in the last decimal place.
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