Trigonometry Calculator

sin θ is the y-coordinate on the unit circle, cos θ the x-coordinate, tan θ the slope

Take the unit circle: radius 1, centered at the origin. Rotate the radius counterclockwise from the positive x-axis by an angle θ and call its endpoint P(x, y). Then sin θ = y, cos θ = x and tan θ = y/x. For 30°, P = (√3/2, 1/2), so sin 30° = 1/2, cos 30° = √3/2 ≈ 0.866025 and tan 30° = (1/2) ÷ (√3/2) = √3/3 ≈ 0.57735. The other three functions are reciprocals: csc θ = 1/sin θ (also written cosec), sec θ = 1/cos θ and cot θ = cos θ/sin θ.

Type an angle into the calculator above and all six values appear together. For multiples of 15° (0°, 15°, 30°, 45°, 60°, 75°, 90°, 120° and so on) you get the exact value with its square roots next to the decimal; for any other angle you get the decimal. Enter the angle in degrees or radians, and radians can be typed with π, as in 5π/6 or pi/4. In the picture, point P on the unit circle and the unrolled 0° to 360° graph move together at the same height. If you know the value and need the angle, as in sin θ = 1/2, switch to "Value → angle".

When you need a trig value

• Ramps and slopes: a slope is rise ÷ run, and that ratio is tan θ. Section 405.2 of the 2010 ADA Standards for Accessible Design says that ramp runs shall have a running slope not steeper than 1:12. Since tan θ = 1/12, θ = arctan(1/12) ≈ 4.763642°, about 8.33 %. Climbing 6 inches on a 1:12 ramp takes 72 inches of horizontal run, and the sloped surface is 6 ÷ sin 4.763642° ≈ 72.25 inches long.
• Splitting a force or a velocity: a slanted force is split with cos and sin. Pulling with 100 N at 30° above the horizontal gives a horizontal part of 100 × cos 30° ≈ 86.6 N and a vertical part of 100 × sin 30° = 50 N. Physics problems about boxes on inclines or balls thrown at an angle use the same step.
• Height from an angle of elevation: standing 50 ft from a tree and sighting its top 35° above horizontal, the part above your eye level is 50 × tan 35° ≈ 35.0 ft. With only a distance and an angle you can measure a height without climbing.
• Rotating coordinates: rotating the point (x, y) by θ about the origin gives (x cos θ − y sin θ, x sin θ + y cos θ). Turning (3, 0) by 30° gives (3 × √3/2, 3 × 1/2) ≈ (2.598, 1.5). Game and graphics code uses this to turn objects or to place points evenly around a circle.
• Checking homework and exam answers: angles such as 150°, 210° or −30° go past 90° or are negative, so the sign must come from the quadrant and the reference angle, and that is where slips happen. The calculator's "Steps" list shows the order: reduce to one turn, quadrant, reference angle, sign.

How to use it

1. At the top choose "Angle → value" or "Value → angle", then the angle unit (degrees or radians) and the number of decimals (4, 6 or 10). Choose 4 decimals to see numbers as in a school trig table.
2. Type the angle in the "Angle θ" box: 150, -30 or 22.5 in degrees; 1.2, 5π/6, pi/4 or 5/6π in radians. In radian mode a π button appears next to the box. The buttons for common angles work too.
3. Read the values on the six cards. Press a card and the diagram shows that function as a segment on the unit circle and as a curve on the unrolled graph. Where a function has no value, the card says "Undefined" with the reason (x = 0 or y = 0, so it can't be divided).
4. Drag point P or click the graph to change the angle. Within 2° of a multiple of 15° it snaps to that angle, and after selecting P the arrow keys move 1° and Page Up and Page Down move 15°.
5. In "Value → angle" pick sin, cos or tan and enter a value such as 1/2, √3/2, −√2/2, 1/√3 or 0.3 (you may type sqrt in place of √). You get the principal value, the solutions with 0° ≤ θ < 360° and the general solution, and a solution button opens that angle in "Angle → value".
6. "Copy result" copies the six values (or the solutions) as text, "Copy link" copies an address that holds your current input, and "Save image" saves the unit circle and the six values as a PNG.

How it is calculated

Definition: the unit circle and six functions

For the point P(x, y) on the unit circle, sin θ = y and cos θ = x; the other four are defined from these two: tan θ = sin θ/cos θ, csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = cos θ/sin θ (NIST DLMF 4.14.4–4.14.7). So tan and sec are undefined at 90° and 270°, where cos θ = 0, and cot and csc are undefined at 0° and 180°, where sin θ = 0. For angles between 0° and 90° these agree with the right-triangle ratios: in a triangle with hypotenuse 1 the height is y and the base is x, so sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent.

Degrees and radians

One full turn is 360°, which is 2π rad. So 180° = π rad, 1° = π/180 rad ≈ 0.017453 rad and 1 rad = 180°/π ≈ 57.29578°. 150° is 150 × π/180 = 5π/6 rad, and 5π/6 rad is 5/6 × 180° = 150°. When you enter radians written with π, the calculator converts them to degrees as exact fractions, so 7π/6 is exactly 210°; radians typed without π, such as 1.2, are treated as decimal approximations.

Quadrant, reference angle and sign

1. Reduce to one turn: adding or subtracting 360° leaves P where it was. −30° gives the same values as 330°, and 750° the same as 30°.
2. Quadrant and reference angle: the reference angle is the acute angle between OP and the x-axis. In quadrant II it is 180° − θ, in quadrant III it is θ − 180° and in quadrant IV it is 360° − θ.
3. Sign: it follows from the signs of x and y. In quadrant I all six functions are positive; in quadrant II only sin and csc; in quadrant III only tan and cot; in quadrant IV only cos and sec.
4. Value: take the value at the reference angle and attach the sign. 210° is in quadrant III with reference angle 30°, so sin 210° = −sin 30° = −1/2, cos 210° = −cos 30° = −√3/2 and tan 210° = tan 30° = √3/3.

Exact values at 15° and 75°

Because 15° = 45° − 30°, the subtraction formula sin(α − β) = sin α cos β − cos α sin β gives sin 15° = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4 ≈ 0.258819. In the same way cos 15° = (√6 + √2)/4 and tan 15° = 2 − √3, and at 75° sin and cos swap, as do tan and cot. The calculator keeps these values, together with those of 30°, 45° and 60°, in a table and uses them directly whenever the reference angle is a multiple of 15°. A value such as tan 30° = 1/√3 is written with a rationalized denominator, √3/3.

Precision of decimal values

For angles that are not multiples of 15°, sin, cos and tan of the reference angle are computed in double-precision floating point and then rounded to the chosen number of places. Working from the reference angle means that symmetric angles such as 150° and 30° give identical decimals for sin. A value that ends exactly (0.5, 1) is shown with "=", and a rounded one with "≈".

Table of special angles (0°–360°)

Two angles that add up to 90° swap sin with cos and tan with cot, as in sin 30° = cos 60° and tan 15° = cot 75°. This follows from sin(90° − θ) = cos θ.

Trig table (0°–90°, every 5°)

For 1° steps or six or more decimals, enter the angle in the calculator above and choose the number of decimals. As the angle approaches 90°, tan grows quickly: 11.4301 at 85° and about 57.29 at 89°.

Finding the angle from a value (inverse functions)

The θ with sin θ = 1/2 is not unique. On the unit circle, points at height 1/2 sit at both 30° and 150°, and every 360° brings you back to the same place. So the calculator first finds one value in a fixed range (the principal value) and then uses symmetry to find the other solutions.

• Range of the principal value: arcsin returns a value from −90° to 90°, arccos from 0° to 180°, and arctan a value greater than −90° and less than 90°. NIST DLMF Table 4.23.1 agrees at the end points: arcsin(±1) = ±π/2, arccos(−1) = π and arccos 1 = 0.
• The second solution: for sin it is 180° − θ, for cos it is 360° − θ, and for tan it is θ + 180°.
• General solution: add 360°n to each solution for sin and cos, and 180°n for tan (n is an integer). In radians that is 2πn and πn.

Sine and cosine only take values from −1 to 1, so an equation like sin θ = 1.2 has no real solution. To find an angle from a csc, sec or cot value, take the reciprocal and enter it as sin, cos or tan: csc θ = 2 means sin θ = 1/2.

Common mistakes

• The calculator or spreadsheet is working in radians: Excel's SIN takes radians. =SIN(30) is the sine of 30 rad, about −0.988; for the sine of 30° you need =SIN(30*PI()/180) or =SIN(RADIANS(30)), which gives 0.5. A scientific calculator does the same if it is in the wrong DEG or RAD mode.
• Writing tan 90° as a huge number: tan 90° would need a division by cos 90° = 0, so it is undefined. In JavaScript, the tangent at the decimal approximation of π/2 comes out as 16331239353195370, and the cosine there is 6.123233995736766 × 10⁻¹⁷ instead of 0. That comes from the approximation of π, not from the mathematics. This calculator recognizes angles exactly on an axis, such as 90° and 270°, as undefined.
• Reading sin⁻¹ as 1/sin: sin⁻¹ x means arcsin x, the angle whose sine is x. The reciprocal 1/sin x is csc x. Likewise sin²θ means (sin θ)², while sin θ² means sin(θ²), which is different.
• Giving only one solution: for sin θ = 1/2 many people write 30° and forget 150°. Check the range the problem asks for (0° ≤ θ < 360°, 0 ≤ θ < 2π and so on) and look for the symmetric second solution.
• Forgetting the sign of the quadrant: writing cos 120° as 1/2, the same as cos 60°, is wrong. 120° is in quadrant II, where the x-coordinate is negative, so cos 120° = −1/2.

Limits

• Exact values with square roots are shown only when the reference angle is a multiple of 15°. Angles such as 18°, 36° or 72°, which do have other exact forms, appear as decimals only.
• Radians typed without π (such as 1.2) and angles that are not multiples of 15° are double-precision approximations. They can be trusted to 10 decimals but not further.
• Angles are calculated only within ±1,000,000°. Radians are held to the same range of angles.
• "Value → angle" handles only sin, cos and tan and finds real solutions only. Complex angles are not calculated.
• What you enter is calculated in your browser and sent nowhere. "Copy link" puts your input into the address.

Basis: the definitions of the trigonometric functions and of the angle units do not change from year to year. The ADA ramp slope was checked on October 20, 2026 in the text of the 2010 ADA Standards published on ada.gov, and the mathematical definitions and the Excel function description were checked in their original sources the same day.

Sources

• 01NIST DLMF §4.14 — Trigonometric Functions: Definitions and Periodicity4.14.4 tan = sin/cos, 4.14.5 csc = 1/sin, 4.14.6 sec = 1/cos, 4.14.7 cot = cos/sin = 1/tan
• 02NIST DLMF §4.23 — Inverse Trigonometric Functions4.23(ii) principal values, Table 4.23.1: arcsin(±1) = ±π/2, arccos(−1) = π, arccos 1 = 0, arctan(±∞) = ±π/2
• 03Wolfram MathWorld — Trigonometry Angles: Pi/12sin(π/12) = (√6 − √2)/4, cos(π/12) = (√6 + √2)/4, tan(π/12) = 2 − √3, csc = √6 + √2, sec = √6 − √2
• 04Wolfram MathWorld — Inverse Sinesin⁻¹ is the same as arcsin; the superscript −1 denotes the inverse function, not the reciprocal
• 05BIPM — The International System of Units (SI Brochure, 9th edition)Table 8: 1° = (π/180) rad
• 06Microsoft Support — SIN functionfor an angle in degrees, multiply by PI()/180 or use the RADIANS function; =SIN(30*PI()/180) = 0.5
• 07ADA.gov — 2010 ADA Standards for Accessible Design405.2 Slope: ramp runs shall have a running slope not steeper than 1:12 (existing sites may use steeper slopes under Table 405.2 where space is limited); confirmed 2026-10-20

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