What this trigonometry calculator does
Trigonometry is the branch of mathematics that connects angles to ratios of sides. Every calculation in the subject — solving a triangle, evaluating sin 30°, converting degrees to radians, or tracing a sine wave — is built on the same handful of definitions. This tool collects all of them into one place and shows the working, not just the answer.
It is organised as ten complete calculators, each covering a different part of the subject, so you can stay on one page instead of opening ten tabs from ten different sites.
- Evaluate — compute all six trigonometric functions for any angle. Enter degrees, radians, gradians, a pi expression like
π/6, or a degrees-minutes-seconds value. Get an exact value when the angle is a special angle, a decimal otherwise, and a note when the function is undefined. The animated unit circle lets you drag, click, use arrow keys, or press Play to watch the point revolve while a live sine wave traces its y-coordinate. - Inverse trig — solve for an angle given a ratio: arcsin, arccos, arctan, arccot, arcsec, arccsc. Handles domain validation, shows the principal value in degrees and radians, plots every solution on a fresh unit circle, and gives the general formula that covers every possible angle.
- Right triangle — enter exactly two of the five values (sides a, b, c, angles A, B) and the solver returns every missing piece plus area, perimeter, altitude to the hypotenuse, circumradius and inradius. Includes a scaled drawing and full SOHCAHTOA steps.
- Oblique triangle — solve any non-right triangle using SSS, SAS, ASA, AAS or SSA. Uses the law of cosines for SSS and SAS, the law of sines for ASA and AAS, and detects the ambiguous SSA case with both solutions drawn side by side.
- Graph — plot
y = A·sin(B(x−C)) + D(or cos/tan) with live amplitude, frequency, phase-shift and vertical-shift sliders. Overlay sin and cos on the same axes, or switch to polar mode to plot roses, cardioids and other classic polar curves. - Identities — type any two expressions and the verifier checks them numerically at hundreds of angles to determine whether they form an identity. Also lists the standard identity families (Pythagorean, sum/difference, double-angle, half-angle, product-to-sum, sum-to-product) as presets.
- Equations — solve
sin(θ) = k,cos(θ) = k, ortan(θ) = kfor every solution on any interval. Handles equality, inequality, and yields the general formula behind the numeric solutions. - Convert — a pure angle-unit converter between degrees, radians, gradians, turns and degrees-minutes-seconds. Type in any cell and every other cell updates instantly.
- Reference — a table of exact values for all 16 special angles, an ASTC quadrant diagram, a formula reference covering every identity you will encounter, and a library of classic right-triangle triples.
- Worksheet — random practice problems across all topics, with hidden answers you can reveal individually or all at once, and a print button for paper practice.
All ten modes share the same visual language and the same underlying math library. The angle you evaluate appears as a marker on the graph, the reference tab mirrors what the tool recognises as special, and the converter updates in lockstep with everything else.
The six trigonometric functions — SOHCAHTOA
In a right triangle with an acute angle θ, name the three sides relative to θ:
- Opposite — the side directly across from θ.
- Adjacent — the side next to θ (not the hypotenuse).
- Hypotenuse — the longest side, opposite the right angle.
The three basic ratios are:
The mnemonic SOHCAHTOA — Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent — is the single most useful thing to memorise in trigonometry. It works for any right triangle, any scale, any unit.
The three reciprocal functions are defined by inverting the three basic ones:
These reciprocals are not the same as the inverse trig functions (arcsin, arccos, arctan). sin⁻¹(x) means "the angle whose sine is x" — it is not 1/sin(x). The notation is unfortunate but universal, and this calculator uses arcsin to avoid the ambiguity.
Degrees vs radians vs gradians vs turns
Angles are measured in units that describe exactly the same thing in different scales. There are four in common use:
- Degrees — a full circle is 360°. The 360 comes from ancient Babylonian astronomy, which used a base-60 numbering system.
- Radians — a full circle is 2π radians. One radian is defined as the angle subtended by an arc equal in length to the radius. Radians are the natural unit for calculus.
- Gradians (also called gons) — a full circle is 400 grad. Historically used in surveying.
- Turns (also called revolutions) — a full circle is 1 turn. Used in engineering (RPM) and computer graphics.
All four agree that a full turn is a full turn, which gives the conversion factors:
So 180° = π radians = 200 grad = 0.5 turn, 90° = π/2 = 100 grad = 0.25 turn, and 1° = π/180 ≈ 0.01745 rad.
The Evaluate tab accepts all four units — just toggle the switch. In radians mode you can type expressions such as pi/6, 3pi/4, 2*pi/3, or 1.5pi and the parser evaluates them.
For measurements written in degrees, minutes and seconds (30° 15′ 45″), use the DMS mode in the Evaluate tab or the DMS field in the Converter tab. One degree is 60 minutes, one minute is 60 seconds — so 30° 15′ 45″ becomes 30 + 15/60 + 45/3600 = 30.2625°.
Exact values for special angles
Most angles have irrational trigonometric values that can only be approximated. But a small set — the special angles — have clean exact values that come from just two triangles.
The 45-45-90 triangle
A right triangle with two equal 45° angles has sides in the ratio 1 : 1 : √2. From it you get:
The 30-60-90 triangle
A right triangle with a 30° angle has sides in the ratio 1 : √3 : 2, where 1 is opposite the 30°, √3 is opposite the 60°, and 2 is the hypotenuse. From it you get:
The remaining special angles (90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°) come from reflecting those values across the unit circle with the correct sign from ASTC. The Reference tab lists all 17 rows.
💡 How this calculator helps. Type 30 in the Evaluate tab and you get sin θ = 0.5 plus the exact value 1/2. Type 210 and you get −0.5 plus −1/2. When the angle is not special, only the decimal appears — the tool will not fabricate an exact form that does not exist.
The unit circle — where the functions really come from
The right-triangle definitions only work for angles between 0° and 90°. To extend them to any angle, mathematicians use the unit circle: a circle of radius 1 centred at the origin.
For an angle θ measured counter-clockwise from the positive x-axis, the terminal side of the angle meets the unit circle at a point (x, y). By definition:
This one definition generates every trig value on the plane. At 0°, the point is (1, 0), so cos 0° = 1 and sin 0° = 0. At 90°, the point is (0, 1), so cos 90° = 0 and sin 90° = 1. At 180°, the point is (−1, 0), so cos 180° = −1. At 270°, the point is (0, −1), so sin 270° = −1.
The interactive circle in the Evaluate tab draws exactly this: the radius line, the horizontal projection (cos), the vertical projection (sin), the reference angle arc, and the current quadrant label. Drag the point around the circle, use the arrow keys on the circle when it is focused, or press Play to watch it revolve automatically while a sine wave traces its y-coordinate on the panel below.
Quadrant signs — the ASTC rule
Because sin = y and cos = x on the unit circle, the sign of each function depends only on which quadrant the angle lies in:
| Quadrant | Angle range | Positive functions |
|---|---|---|
| I | 0° – 90° | All six |
| II | 90° – 180° | sin, csc only |
| III | 180° – 270° | tan, cot only |
| IV | 270° – 360° | cos, sec only |
The mnemonic is ASTC — All, Sine, Tangent, Cosine — often taught as "All Students Take Calculus". Every other function is negative in that quadrant. The Evaluate tab shows the quadrant and the reference angle automatically, so you can verify the sign manually if you need to.
Inverse trigonometry
An inverse trig function answers the question: given a ratio, what angle produces it? For example, if sin θ = 0.5, then θ = arcsin(0.5) = 30°.
There is a catch. Because sin(θ) = sin(180° − θ) and every trig function is periodic, a single ratio matches infinitely many angles. To make arcsin, arccos and arctan into functions, mathematicians restrict them to a principal branch:
| Function | Domain | Principal range |
|---|---|---|
| arcsin(x) | −1 ≤ x ≤ 1 | −90° to 90° |
| arccos(x) | −1 ≤ x ≤ 1 | 0° to 180° |
| arctan(x) | all real | −90° to 90° |
| arcsec(x), arccsc(x) | |x| ≥ 1 | 0° to 180°, excluding 90° |
| arccot(x) | all real | 0° to 180° |
To recover all solutions, add the period:
- arcsin: θ = α + 360°k or θ = 180° − α + 360°k
- arccos: θ = ±α + 360°k
- arctan: θ = α + 180°k
The Inverse tab shows the principal value, plots every solution on the unit circle, and displays the general-solution formula, so you can see exactly which angles fit the input ratio.
Solving right triangles
Every right triangle has six "parts": three sides (a, b, c) and three angles (A, B, C = 90°). Given enough information, you can always find the rest.
If you know two sides, the third follows from the Pythagorean theorem:
If you know one side and one acute angle, the remaining sides come from SOHCAHTOA. For example, with A and c known: a = c · sin A, b = c · cos A. With A and a known: c = a / sin A, b = a / tan A.
The Right triangle tab accepts any valid pair — two sides, or one side plus one acute angle — and computes everything else. It also gives area (½ab), perimeter (a + b + c), altitude to the hypotenuse (ab/c), circumradius (c/2), and inradius ((a + b − c)/2).
📌 Worked example — the classic 3-4-5. Given a = 3 and b = 4: c = √(9 + 16) = 5. Angle A = arctan(3/4) ≈ 36.87°. Angle B = 90° − 36.87° ≈ 53.13°. Area = ½ · 3 · 4 = 6. Perimeter = 12. The tool draws the triangle to scale and lists each of these values.
Oblique triangles — law of sines and law of cosines
For triangles without a right angle, SOHCAHTOA does not apply. Instead you use the two laws.
Law of sines
Useful when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA).
Law of cosines
Useful when you know all three sides (SSS) or two sides and the included angle (SAS). It is the general form of the Pythagorean theorem — when C = 90°, cos 90° = 0 and the term vanishes, recovering a² + b² = c².
The ambiguous SSA case
The one configuration that can produce more than one triangle is SSA — two sides and a non-included angle. Because sin(θ) = sin(180° − θ), the same inputs can sometimes fit two different triangles. Given sides a, b and angle A (opposite a), let h = b · sin A:
- a < h → no solution (the side is too short to reach the opposite side).
- a = h → exactly one solution (a right triangle).
- h < a < b → two solutions (the ambiguous case).
- a ≥ b → exactly one solution.
The Oblique triangle tab detects this and, when two solutions exist, draws both triangles side by side with all their sides and angles.
Area formulas
Depending on what you know, any of these will do:
Graphing trig functions
A transformed sine function has the general form:
Each parameter has a precise geometric meaning:
- A — amplitude. The curve oscillates between D − |A| and D + |A|. Base sine has amplitude 1.
- B — frequency. Period = 2π / |B| radians (or 360° / |B|). Doubling B halves the period.
- C — phase shift. The curve slides C units to the right (if positive) or left (if negative).
- D — vertical shift. The midline of the wave moves up or down by D.
Cosine is sine shifted left by π/2. Tangent is different: its period is π (not 2π), it has no amplitude because it is unbounded, and it has vertical asymptotes wherever cos(x) = 0 — at 90°, 270°, and every odd multiple of 90°.
The Graph tab draws the curve live as you drag the sliders, and can overlay sine and cosine together so you can see their phase relationship. Switch to polar mode to plot roses, cardioids, and lemniscates — classic curves that only appear when you use polar coordinates.
Trigonometric identities
An identity is an equation that is true for every angle where both sides are defined. Trigonometry has a rich family of them, and they are the basis for simplifying expressions and solving equations.
The most important groups are:
- Pythagorean — sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.
- Quotient — tan θ = sin θ / cos θ, cot θ = cos θ / sin θ.
- Cofunction — sin(90° − θ) = cos θ, and so on.
- Even/odd — sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ.
- Sum and difference — sin(A ± B), cos(A ± B), tan(A ± B).
- Double-angle — sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ, tan 2θ = 2 tan θ / (1 − tan²θ).
- Half-angle — sin(θ/2), cos(θ/2), tan(θ/2).
- Product-to-sum and sum-to-product — for converting products into sums and back.
The Identities tab lets you test any identity numerically by evaluating both sides at hundreds of angles. If they agree at every angle to machine precision, the identity is virtually certain to be true; if even one angle disagrees, it is not an identity.
Solving trig equations
A trig equation like sin(θ) = 0.5 has infinitely many solutions, because the trig functions are periodic. The Equations tab finds every solution on any interval and shows the general formula.
The general solutions are:
- sin θ = k: θ = arcsin(k) + 360°k or θ = 180° − arcsin(k) + 360°k
- cos θ = k: θ = ± arccos(k) + 360°k
- tan θ = k: θ = arctan(k) + 180°k
So sin θ = 0.5 has solutions at 30° and 150° on [0°, 360°), plus every one of those plus 360°k. The Equations tab lists all of them in the chosen interval.
Where trigonometry is actually used
- Surveying and construction. Angles of elevation and depression, measuring heights of buildings and distance across rivers — pure right-triangle trigonometry.
- Navigation and GPS. Great-circle distances, bearings, and trilateration all reduce to spherical trigonometry.
- Engineering. AC circuit analysis uses phasors (rotating vectors), which are trigonometric by construction.
- Physics. Projectile motion, simple harmonic motion, waves, and optics are all written in sines and cosines.
- Computer graphics and games. Rotating an object, aiming a camera, or computing a normal vector requires trig.
- Astronomy. Parallax, angular size, and orbital mechanics use the law of sines and cosines on the celestial sphere.
- Music and acoustics. The Fourier transform decomposes sound into sine waves — the harmonic content of every note.
- Medical imaging. CT and MRI reconstruction rely on the Fourier slice theorem, which is trigonometric at its base.
- Machine learning. Fourier features, positional encodings in transformers, and many kernels rely on sin and cos.
- Cryptography and signal processing. FFT-based algorithms require trigonometric primitives.
Common mistakes
- Using the wrong angle mode. If your calculator is in radians and you type
30, you get sin(30 rad) ≈ −0.988 instead of 0.5. Always check the unit switch. - Confusing sin⁻¹ with csc. sin⁻¹(x) means arcsin — the inverse function — not 1/sin(x).
- Forgetting the negative sign in quadrants II, III and IV. sin 150° is +1/2, not −1/2. Use ASTC or draw the unit circle.
- Assuming tan 90° is a large number. It is undefined — the tangent curve has a vertical asymptote there.
- Using the law of sines on SSA without checking the ambiguous case. You may miss the second triangle.
- Rounding too early. Carry full precision through intermediate steps; round only at the end.
- Mixing degrees and radians in the same expression. All angles in an identity or law must be in the same unit.
- Forgetting that only the principal branch of an inverse trig function is returned by a calculator. The full solution set requires adding the period.
Limitations
- Exact values are only shown for the 16 special angles (multiples of 30° and 45°). Other angles show decimal values only.
- Precision is double-precision floating point (about 15–17 significant digits). Very large angles may show tiny rounding artefacts near special values; the tool uses a 1e-9 tolerance to recognise special angles.
- Right-triangle and oblique-triangle modes assume planar (Euclidean) geometry. They do not solve spherical triangles.
- The graph mode plots up to two Cartesian functions or one polar function at a time. It does not superimpose more than that.
- The identity verifier checks numerically, not symbolically. When two expressions differ by less than machine precision at every sampled angle, the tool concludes they are probably identical — a symbolic proof is beyond its scope.
- The equation solver handles single-function equations of the form
sin(θ) = k,cos(θ) = k,tan(θ) = k. More complex equations can be solved by entering equivalent forms in the identity verifier. - All calculations run in your browser with no server — nothing is uploaded, and nothing persists after you clear your browser's site data.
Frequently asked questions
What is a trigonometry calculator?
A trigonometry calculator evaluates the six trigonometric functions (sin, cos, tan, csc, sec, cot) for an angle, computes inverse trigonometric functions, and solves triangles. This tool combines ten trig calculators in one page: function evaluation with exact special-angle values, inverse trig with general solutions, right triangle solver, oblique triangle solver with the ambiguous case, live graphing with polar mode, identity verification, trig equation solving, unit conversion, reference tables, and a worksheet generator.
What is sin 30 degrees?
sin(30°) = 1/2 = 0.5 exactly. 30° is a special angle in the 30-60-90 triangle, where the side opposite 30° is half the hypotenuse. In radians, 30° = π/6.
What is cos 60 degrees?
cos(60°) = 1/2 = 0.5 exactly. In the 30-60-90 triangle, the side adjacent to 60° is half the hypotenuse. In radians, 60° = π/3.
What is tan 45 degrees?
tan(45°) = 1 exactly. In the 45-45-90 triangle, both legs are equal, so opposite/adjacent = 1. In radians, 45° = π/4.
What is sin 45 degrees?
sin(45°) = √2/2 ≈ 0.7071. 45° is the angle in the isosceles right triangle with sides 1, 1, √2.
What is cos 30 degrees?
cos(30°) = √3/2 ≈ 0.8660. It comes from the 30-60-90 triangle where the adjacent side (√3) is over the hypotenuse (2).
What is tan 60 degrees?
tan(60°) = √3 ≈ 1.7321. In the 30-60-90 triangle, the side opposite 60° is √3 and the adjacent is 1, giving √3/1.
What is sin 90 degrees?
sin(90°) = 1 exactly. At 90° the point on the unit circle reaches the top at (0, 1), so sine (the y-coordinate) is 1.
What is cos 0 degrees?
cos(0°) = 1 exactly. At 0° the point on the unit circle is at (1, 0), so cosine (the x-coordinate) is 1 and sine is 0.
Why is tan 90 degrees undefined?
tan(90°) is undefined because tan(θ) = sin(θ)/cos(θ), and cos(90°) = 0. Dividing by zero has no defined value, so the tangent function has a vertical asymptote at 90°, 270° and every odd multiple of 90°. This calculator reports "undefined" rather than returning a wrong number.
Why is sec 90 degrees undefined?
sec(θ) = 1/cos(θ). Because cos(90°) = 0, sec(90°) requires dividing by zero, which is undefined. The same is true at 270° and every odd multiple of 90°.
Why is csc 0 degrees undefined?
csc(θ) = 1/sin(θ). Because sin(0°) = 0, csc(0°) requires dividing by zero, which is undefined. The same is true at 180°, 360° and every multiple of 180°.
How do I convert degrees to radians?
Multiply degrees by π/180. For example, 30° × π/180 = π/6 radians. In general: radians = degrees × π / 180. The conversion is exact because 180° = π radians by definition.
How do I convert radians to degrees?
Multiply radians by 180/π. For example, π/4 × 180/π = 45°. A useful approximation: 1 radian ≈ 57.2958°.
How do I convert degrees to gradians?
Multiply degrees by 10/9. A full circle is 400 gradians (also called gons), so 90° = 100 grad, 180° = 200 grad, 360° = 400 grad. The Converter tab handles this and reverse conversion automatically.
How do I convert degrees to turns?
Divide degrees by 360. A full circle is 1 turn (also called 1 revolution). So 180° = 0.5 turn, 90° = 0.25 turn, and 1° = 1/360 turn. Used in engineering for RPM and in computer graphics for rotation angles.
What is SOHCAHTOA?
SOHCAHTOA is a mnemonic for the three basic trig ratios in a right triangle. SOH: Sine = Opposite / Hypotenuse. CAH: Cosine = Adjacent / Hypotenuse. TOA: Tangent = Opposite / Adjacent. In a right triangle with angle θ, "opposite" is the side facing θ, "adjacent" is the side next to θ, and "hypotenuse" is the longest side.
How do I find sin cos tan without a calculator?
For the special angles 0°, 30°, 45°, 60° and 90° you can derive exact values from two triangles: the 45-45-90 (sides 1, 1, √2) and the 30-60-90 (sides 1, √3, 2). For any other angle, exact values require infinite series or half-angle identities, which is why calculators are normally used. This tool shows the exact value automatically whenever the angle is a special angle.
What are the exact values of sin cos tan?
For the primary special angles: sin(0°)=0, sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1. cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0. tan(0°)=0, tan(30°)=√3/3, tan(45°)=1, tan(60°)=√3, tan(90°) is undefined. The Reference tab lists all 17 special angles around the unit circle.
What is the unit circle?
The unit circle is a circle of radius 1 centred at the origin. For any angle θ measured counter-clockwise from the positive x-axis, the point where the terminal side of θ meets the circle has coordinates (cos θ, sin θ). That is the geometric definition of sine and cosine, and it extends the trig functions to all angles — not just acute angles inside a right triangle.
How do I use the animated unit circle?
Drag the point around the circle, click anywhere on the circle, focus the circle and use the arrow keys (arrow = 1°, Shift+arrow = 15°), or press the Play button to watch it revolve automatically. The angle, sine, cosine, tangent, quadrant and reference angle update live in the panel next to the circle, and a parallel sine wave traces the y-coordinate as the point revolves, showing graphically how sine is generated.
What is the ASTC rule?
ASTC ("All Students Take Calculus") is a mnemonic for the signs of the trig functions in each quadrant. Quadrant I: all six functions positive. Quadrant II: only sine (and cosecant) positive. Quadrant III: only tangent (and cotangent) positive. Quadrant IV: only cosine (and secant) positive.
What is a reference angle?
The reference angle is the acute angle between the terminal side of θ and the closest x-axis. In quadrant I it is θ itself. In QII it is 180° − θ. In QIII it is θ − 180°. In QIV it is 360° − θ. Reference angles let you reduce any trig value to a first-quadrant value plus the correct sign from ASTC.
What is an inverse trig function?
An inverse trig function (arcsin, arccos, arctan) answers the question "what angle has this sine/cosine/tangent value?" Because the trig functions are periodic, the inverse is restricted to a principal branch: arcsin returns −90° to 90°, arccos returns 0° to 180°, arctan returns −90° to 90°. The general solution adds a period (2πk for sin/cos, πk for tan) to cover all possible angles. The Inverse tab shows the principal value plus plots every solution on the unit circle.
What is the domain of arcsin and arccos?
Both arcsin(x) and arccos(x) require x to be between −1 and 1 inclusive, because sine and cosine only ever output values in that range. arctan accepts any real number. arcsec(x) and arccsc(x) require |x| ≥ 1.
How do I solve a right triangle?
You need exactly two known values, and at least one of them must be a side. In this calculator, enter any two of a, b, c, angle A or angle B. The solver uses the Pythagorean theorem (a² + b² = c²) plus SOHCAHTOA to find the rest. For example, given a = 3 and b = 4: c = 5, angle A = arctan(3/4) ≈ 36.87°, angle B ≈ 53.13°.
What is the 30-60-90 triangle?
A 30-60-90 triangle is a right triangle with angles 30°, 60° and 90°. Its sides are always in the ratio 1 : √3 : 2, where 1 is opposite the 30° angle, √3 is opposite the 60° angle, and 2 is the hypotenuse. This triangle is the source of the exact sin and cos values for 30° and 60°.
What is the 45-45-90 triangle?
A 45-45-90 triangle is a right triangle with two equal 45° angles. Its sides are always in the ratio 1 : 1 : √2, where √2 is the hypotenuse. It is the source of the exact sin and cos values for 45° (both √2/2) and tan 45° = 1.
What are the special right triangle triples?
The most useful Pythagorean triples for trigonometry are the 3-4-5 (angles ≈ 36.87° and 53.13°), the 5-12-13 (≈ 22.62° and 67.38°), the 8-15-17 (≈ 28.07° and 61.93°), and the 7-24-25 (≈ 16.26° and 73.74°). The 30-60-90 and 45-45-90 triangles are the only right triangles whose side ratios involve irrationals only in √2 or √3 — all these are recognised by the Right triangle solver.
What is the law of sines?
The law of sines states that a / sin A = b / sin B = c / sin C = 2R, where a, b, c are the sides, A, B, C are the opposite angles, and R is the circumradius. It works for any triangle and is most useful when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA).
What is the law of cosines?
The law of cosines states that c² = a² + b² − 2ab·cos(C), and analogously for the other sides. It is the general version of the Pythagorean theorem, with the −2ab·cos(C) term correcting for the triangle not being right-angled. It is most useful for SSS (all three sides) and SAS (two sides and the included angle).
What is the ambiguous case of the law of sines?
The ambiguous case is the SSA configuration — two sides and a non-included angle. Because sin(θ) = sin(180° − θ), there can be two different triangles that fit the same SSA data. Depending on the relationship between the given side and the height of the triangle (h = b·sin A), you get zero, one or two solutions. This calculator detects and displays both solutions when two exist, with a side-by-side visual comparison.
What is the area of a triangle using trigonometry?
There are three common area formulas. If you know two sides and the included angle: Area = ½ · a · b · sin(C). If you know all three sides, use Heron's formula: Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) / 2. For a right triangle, Area = ½ · a · b. This calculator uses whichever formula fits the inputs.
How do I graph sine and cosine functions?
For a function of the form y = A·sin(B(x − C)) + D, the amplitude is |A|, the period is 2π/|B| (or 360/|B| in degrees), the phase shift is C (right if positive), and the midline is y = D. The cosine graph is the sine graph shifted left by π/2. Tangent has period π/|B| (180/|B|) rather than 2π and has vertical asymptotes wherever cos(x) = 0. The Graph tab draws this live as you move the sliders, and can overlay sine and cosine on the same axes.
What is the period of sin and cos?
The period of sin(x) and cos(x) is 2π radians, or 360°. After that interval the function repeats. For a transformed function y = sin(Bx), the period is 2π/|B|, so a factor of B = 2 halves the period to π. Tangent has period π radians, or 180°.
What is the amplitude of a trig function?
The amplitude is half the distance between the maximum and minimum values. For y = A·sin(x), the amplitude is |A|. The base sine and cosine functions have amplitude 1 (they oscillate between −1 and 1). Tangent has no amplitude because it is unbounded.
What are polar graphs?
A polar graph plots a curve where the coordinates are (r, θ) instead of (x, y). You specify a radius that depends on the angle, and the shape traces out. Classic polar curves include the rose r = a·cos(kθ) (k petals if k is odd, 2k petals if even), the cardioid r = a(1 + cos θ), the lemniscate r² = a²·cos(2θ), and the spiral r = aθ. The Graph tab includes a polar plotter for these.
How do I verify a trig identity?
Enter the left and right sides of any trig identity as expressions, and the Identities tab will test them numerically at many random angles and symbolically evaluate the difference. If every value matches to machine precision, the identity is likely true; if any angle gives a different result, it is not an identity. The tab also lists the standard identities (Pythagorean, sum/difference, double-angle, half-angle, product-to-sum, sum-to-product) for reference.
How do I solve a trig equation?
Enter an equation of the form sin(x) = value, cos(x) = value, tan(x) = value. The Equations tab finds all solutions on any interval, using the principal value plus the period. For sin(x) = k, the general solution is x = arcsin(k) + 360°k or x = 180° − arcsin(k) + 360°k. For cos(x) = k, x = ±arccos(k) + 360°k. For tan(x) = k, x = arctan(k) + 180°k. This tool lists them numerically on your interval.
What are the reciprocal trig functions?
The three reciprocal trig functions are cosecant, secant and cotangent: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). Each has the same sign as its underlying function in every quadrant, and each is undefined exactly where its underlying function is zero.
What are the Pythagorean trig identities?
The three Pythagorean identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. They follow directly from the unit circle: since sin θ = y and cos θ = x, and x² + y² = 1 on the unit circle, sin²θ + cos²θ = 1. The other two are derived by dividing that identity by cos²θ and sin²θ respectively.
What are the sum and difference formulas?
sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, and tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B). These formulas generate the double-angle and half-angle identities and are the basis for trigonometric addition theorems.
What are the double-angle formulas?
sin 2θ = 2 sin θ cos θ. cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ (three equivalent forms, choose whichever is convenient). tan 2θ = 2 tan θ / (1 − tan²θ).
What are the half-angle formulas?
sin(θ/2) = ±√((1 − cos θ)/2). cos(θ/2) = ±√((1 + cos θ)/2). tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ. The ± sign is chosen based on which quadrant θ/2 is in.
What are the product-to-sum formulas?
sin A · sin B = ½[cos(A − B) − cos(A + B)]. cos A · cos B = ½[cos(A − B) + cos(A + B)]. sin A · cos B = ½[sin(A + B) + sin(A − B)]. These convert products into sums, which is useful for integration and Fourier analysis.
What are the sum-to-product formulas?
sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2). sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2). cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2). cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2).
What is the law of tangents?
(a − b)/(a + b) = tan((A − B)/2) / tan((A + B)/2). It is useful for solving triangles when you know two sides and their opposite angles, but is less commonly used than the law of sines and cosines.
What is radians in terms of arc length?
One radian is the angle subtended by an arc whose length equals the radius. For a circle of radius r, an arc of length s subtends an angle of θ = s/r radians. This is why radians are the natural unit for calculus — d/dx sin x = cos x only when x is in radians.
Why is 1 radian about 57.3°?
A full circle is 2π radians or 360°, so 1 radian = 360/(2π) = 180/π ≈ 57.2958°. This comes from the definition of π as the circumference of a circle divided by its diameter.
What is the difference between arcsin and sin⁻¹?
They are the same function — sin⁻¹(x) means the inverse sine, also written arcsin(x). It is NOT the same as (sin(x))⁻¹, which is csc(x) = 1/sin(x). The superscript −1 on a function name means 'inverse function', but on a value it means 'reciprocal'. This calculator uses arcsin notation to avoid the ambiguity.
What are the units of the triangle solver?
The triangle solvers are unit-agnostic — enter all sides in the same unit (centimetres, inches, metres, feet — anything), and the results are in that same unit. Areas are in the square of that unit. Angles are always in degrees unless the tool explicitly states otherwise.
Does the oblique triangle solver handle all configurations?
Yes. It handles SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle, including the ambiguous case with two solutions). This covers every valid combination of the minimum three given values.
Why does the same angle sometimes show a different tangent?
Tangent is periodic with period 180°, so tan(θ) = tan(θ + 180°). The value is the same but the line goes in the opposite direction on the unit circle. Also, tangent has vertical asymptotes at 90°, 270° and every odd multiple of 90°, where it is undefined — the tool reports "undefined" rather than a very large number.
Can I use this calculator for trigonometry homework?
Yes. Every mode shows the step-by-step working where the steps are informative: the Evaluate tab shows exact values and quadrant/reference-angle reasoning, the right triangle tab shows the SOHCAHTOA steps, the oblique triangle tab shows which law was applied, and the Equations tab shows the general-solution reasoning. That makes it useful for learning, not just checking answers.
Is this calculator accurate?
Yes. It uses the standard JavaScript Math library, which is accurate to double-precision floating point (about 15–17 significant digits). Exact values are detected for all 16 special angles around the unit circle. There is a small floating-point tolerance (about 1e-9) around special angles so that typing 30.0000000001° still recognises sin = 1/2.
Does the Worksheet tab generate practice problems?
Yes. The Worksheet tab generates randomised practice problems covering evaluation of special angles, exact values, triangle solving, inverse trig and identity verification. Each problem has a hidden answer you can reveal individually or all at once, and there is a Print button to create a physical worksheet.
What does the Convert tab do that the Evaluate tab does not?
The Evaluate tab focuses on one angle at a time and shows all six trig functions. The Convert tab is a pure angle-unit converter: it converts between degrees, radians, gradians, turns and DMS in a single grid, so you can see one angle in all five representations at once. It also has a reverse converter — type in any cell and every other cell updates.
Can I export the results?
Yes. Every tab has export options: copy to clipboard, download as CSV (for the tables), and download the visualizations as PNG (for the graphs and the unit circle). The Share button encodes the full state into the URL hash so you can bookmark or share a specific configuration.
Does the tool save my inputs?
Yes, recent evaluations, triangle solutions and worksheet problems are stored in your browser's localStorage so you can restore them. Nothing is uploaded or logged. You can clear the history at any time.
Is my data private?
Yes. Everything runs entirely in your browser — all calculations use JavaScript's built-in Math library. There are no network requests, no analytics, no accounts and no server involved. The Share button encodes the current state into the URL hash on your own device only.
How is this different from other trigonometry calculators?
Most trig calculators focus on one thing: a function evaluator, a triangle solver, or a grapher. This tool combines ten separate calculators in one page, with shared state (the angle you enter in Evaluate appears as a marker on the graph), exact values where available, step-by-step working, animated visualisations, an identity verifier, a trig equation solver with general solutions, a worksheet generator and a comprehensive reference library. All calculations run client-side with no server dependency.
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Sources & methodology
- NIST Guide for the Use of the International System of Units (SI) — official definition of radian as an SI derived unit, and the 180° = π radian relationship.
- Wolfram MathWorld — Trigonometry — definitive definitions of the six trigonometric functions, exact special-angle values, identities and the law of sines/cosines.
- OpenStax Precalculus 2e, Chapter 5 — Unit Circle — open-access textbook coverage of the unit-circle definition of sine and cosine.
- OpenStax Precalculus 2e, Chapter 8 — Law of Sines and Law of Cosines — including the SSA ambiguous case conditions.
- MDN Web Docs — JavaScript Math object — the standard trigonometric functions used by this calculator.
- Wikipedia — Trigonometric functions — historical context, exact values for 15°, 18°, 72°, 75°, and principal-value definitions of the inverse functions.
- Wikipedia — Law of sines and Law of cosines — full derivations and the ambiguous-case conditions.
- Wikipedia — List of trigonometric identities — the full catalogue of Pythagorean, sum/difference, double-angle, half-angle, product-to-sum and sum-to-product identities.
- Wikipedia — Polar coordinate system — including the polar roses, cardioids, lemniscates and spirals plotted by the Graph tab.