Sin Calculator: Free Trigonometry Tool Online
A sin calculator should do more than one ratio. This one computes sin in degrees, radians or gradians, reverses it with arcsin, evaluates all six trigonometric functions at once, and applies the law of sines to triangle problems, with quadrant detection and a full breakdown of every result.
| Degrees | Radians | sin | cos | tan |
|---|
Quick Answer
The pre-filled run above computes sin(30°) = 0.5, with cos = 0.8660254038 and tan = 0.5773502692, sitting in Quadrant I with a reference angle of 30°. The calculator runs four modes from one page: plain sine, arcsin, all six trig functions, and law-of-sines triangle solving, in degrees, radians or gradians.
Key takeaways
- Sine is the y-coordinate of the unit-circle point at your angle: sin(30°) = 0.5, sin(90°) = 1, sin(270°) = -1, and it never leaves the range -1 to 1.
- Check the unit before you trust any sine value. Spreadsheets and code take radians, so =SIN(30) in a spreadsheet computes the sine of 30 radians, not 30 degrees.
- Arcsin reverses the question: arcsin(0.5) = 30°, the principal value between -90° and 90°.
- The law of sines solves a whole triangle from one side-angle pair: a/sin(A) = b/sin(B) = c/sin(C).
- Quick Answer
- How to use the sin calculator
- How the sine math works
- Worked example: sin(30°), arcsin(0.5) and a triangle
- Quadrants, reference angles and the unit circle
- Exact values worth memorizing
- The law of sines, explained
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the sin calculator
Pick the mode that matches your question, keep the worked-example defaults or type your own numbers, and press Calculate. Results stay hidden until you click.
- sin(θ) mode: enter the angle and choose its unit. Try 30 degrees, or the same angle as 0.5236 radians, or 33.3333 gradians: all three return 0.5.
- arcsin mode: enter a value between -1 and 1, for example 0.5, and read the angle back in degrees or radians.
- All functions mode: one angle gives you sin, cos, tan, csc, sec and cot together, plus the quadrant badge and the identity check sin²(θ) + cos²(θ) = 1.
- Law of sines mode: enter side a and angle A, then solve for angle B from a known side b, or for side b from a known angle B.
Two honest details about how this tool behaves. First, at exactly 90° or 270° the main result prints tan as “Undefined”, because cos is zero there and tan has no finite value. Floating-point arithmetic never produces exactly zero for cos(90°), which used to print the artifact 16331239353195370; the tool now treats any computed tan magnitude above 10¹⁰ as an asymptote, the same convention the reference table below the results uses. Second, in the ambiguous side-side-angle case the solver returns the principal value of angle B only. If you need cosine as the starting point instead, the cosine calculator runs the same four modes for the adjacent ratio.
How the sine math works
In a right triangle, sine is opposite over hypotenuse. On the unit circle it generalizes to every angle: sin(θ) is the y-coordinate of the point reached by rotating θ from the positive x-axis. That is why sin(150°) = 0.5 (Quadrant II, positive) while sin(210°) = -0.5 (Quadrant III, negative), and why sine repeats every 360°.
Radians = Degrees × π ÷ 180 · Gradians = Degrees × 400 ÷ 360
Reciprocals
csc(θ) = 1 ÷ sin(θ) · sec(θ) = 1 ÷ cos(θ) · cot(θ) = 1 ÷ tan(θ)
Identity (checked on every result)
sin²(θ) + cos²(θ) = 1
Law of sines
a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C)
The unit-circle definition is the one every serious reference uses, from OpenStax Precalculus to Paul’s Online Math Notes at Lamar University: (cos θ, sin θ) is the point itself, which extends all six functions past the 0°-90° range of right triangles. A central angle calculator covers the circle-geometry side of the same idea when you are working with arcs and sectors.
Worked example: sin(30°), arcsin(0.5) and a triangle
The pre-filled sin run is the classic: 30 degrees gives sin = 0.5, cos = 0.8660254038, tan = 0.5773502692, csc = 2, sec = 1.1547005384, cot = 1.7320508076, in Quadrant I with the identity check reading sin²(θ) + cos²(θ) = 1. Here are the other verified runs worth knowing, each reproducible in the calculator above.
| Run | Main result | Companion values |
|---|---|---|
| sin(30°) | 0.5 | cos 0.8660254038, tan 0.5773502692 |
| sin(45°) (All functions) | 0.7071067812 | csc = sec = 1.4142135624, cot = 1 |
| arcsin(0.5) | 30° | 0.5235987756 rad in radian mode |
| Law of sines: a=10, A=30°, B=45° | b = 14.142136 | C = 105°, c = 19.318517 |
| Law of sines: a=10, A=30°, b=5 | B = 14.477512° | C = 135.522488°, c = 14.012585 |
Read the sign pattern across quadrants and it is hard to misremember: sine is positive where the unit-circle point sits above the x-axis (Quadrants I and II) and negative below it (Quadrants III and IV).
Quadrants, reference angles and the unit circle
A reference angle is the acute angle between the terminal side and the x-axis: for 210° it is 30°, which is why sin(210°) = -sin(30°). The calculator reports both the normalized angle and its reference angle on every run, so a surprising sign is always explainable.
The quadrant badge above the results does this bookkeeping for you. If you want the reference-angle idea as its own tool, the reference angle calculator computes it directly, and the inverse sine calculator is the dedicated arcsin page.
Exact values worth memorizing
Most angles give messy decimals, but the special angles give clean surds. These are the ones textbooks expect you to know, and the reference table inside the calculator shows their decimal forms next to every result.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
The law of sines, explained
For any triangle, side a over sin(A) equals side b over sin(B) equals side c over sin(C). One known side-angle pair fixes that common ratio, and the rest of the triangle follows: b = a · sin(B) / sin(A), and angle C is whatever is left of 180°.
The honest caveat is the ambiguous case. Given side a, angle A and side b, the arcsine step can have two answers: angle B and its supplement 180° – B may both complete a valid triangle. This calculator returns the principal value (the acute one) and notes it, following the standard treatment in OpenStax’s law of sines chapter. If your inputs admit the second triangle, solve it by hand with B′ = 180° – B. The tool also refuses impossible triangles outright: side b too long for the given angle returns “No valid triangle exists with these values (sin B > 1)”, and angle sums at or above 180° are rejected with their own messages.
What competing calculators miss
Most pages ranking for this keyword compute sin and stop, and the detailed ones each stop short in a different place:
- ShortCalc: the strongest rival, with exact surd values and six angle units. It has no arcsin mode and no law of sines, so triangle problems send you elsewhere.
- Calkoo: all six functions plus inverse trig, a unit circle view and a graph, and it accepts pi expressions. It covers no law of sines and no gradians.
- CalculatorLib: a clean single-purpose sine tool, but degrees and radians only, no quadrant or reference-angle readout, and its arcsin page prints “30 rad” where the radian answer is 0.5236. Unit handling matters, and this page gets it right.
- StatGarden and EpicCalculators: solid sin/cos/tan triple tools with formula write-ups, but no inverse mode and no triangle solving.
- The gap this page fills: four modes on one page (sin, arcsin, all six functions, law of sines), gradians alongside degrees and radians, a quadrant badge with reference angle on every result, the sin² + cos² identity check, tan shown as Undefined at the 90° and 270° asymptotes, and open disclosure of the ambiguous SSA case that most law-of-sines tools silently ignore.
Frequently asked questions
In a right triangle, sine is the length of the side opposite the angle divided by the hypotenuse. On the unit circle it is the y-coordinate of the point at your angle, which is why sin(30°) = 0.5, sin(90°) = 1, and sine stays between -1 and 1 for every angle.
Multiply degrees by π/180 for radians and by 400/360 for gradians. A full circle is 360 degrees, 2π radians, or 400 gradians, so 30° = 0.5236 rad = 33.3333 grad, and this calculator converts for you automatically.
Arcsin (sin⁻¹) reverses the sine function: give it a ratio between -1 and 1 and it returns the angle that produced it. Arcsin(0.5) = 30°. Use it when you know two sides of a right triangle and need the angle between them. The answer is the principal value between -90° and 90°.
It does not anymore. Cos(90°) is zero, so tan(90°) has no finite value. Floating-point arithmetic computes cos(90°) as a tiny non-zero value, which used to produce the artifact 16331239353195370; the tool now treats any tan magnitude above 10¹⁰ as an asymptote and prints Undefined, matching the reference table.
It solves a triangle from one known side-angle pair, since a/sin(A) = b/sin(B) = c/sin(C). Give it side a and angle A plus either side b or angle B, and it returns the missing angle or side along with the rest of the triangle. Inputs that cannot form a triangle are rejected with an explanation.
When you know side a, angle A and side b, angle B can have two answers: the principal value and its supplement 180° − B may both complete valid triangles. This calculator returns the principal value and says so; solve the second triangle by hand with the supplementary angle.
Spreadsheets and programming languages take radians, so =SIN(30) computes the sine of 30 radians (about −0.988), not 30 degrees. Use =SIN(RADIANS(30)) for degrees, or enter the radian value directly in this calculator’s radian mode.
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References
- OpenStax. Precalculus 2e: Unit Circle: Sine and Cosine Functions. Accessed October 23, 2026.
- OpenStax. Precalculus 2e: Non-right Triangles: Law of Sines. Accessed October 23, 2026.
- Khan Academy. Trigonometry: Unit circle. Accessed October 23, 2026.
- Paul Dawkins, Lamar University. Paul’s Online Notes: Trig Functions. Accessed October 23, 2026.
- Math is Fun. Sine, Cosine and Tangent. Accessed October 23, 2026.
- National Institute of Standards and Technology. Digital Library of Mathematical Functions: Trigonometric Functions. Accessed October 23, 2026.