Trigonometry Calculator

A trigonometry calculator (trig calculator) finds sin, cos, tan and their inverses for any angle and solves right triangles. Enter 30° and it returns sin 30° = 0.5 (exactly 1/2), cos 30° ≈ 0.8660 (√3/2) and tan 30° ≈ 0.5774 (√3/3). Switch to triangle mode and one angle plus one side gives every side, the second angle, the area and the perimeter.

Trigonometry answers questions about angles and lengths. If you know two sides of a right triangle and no angle, the Pythagorean Theorem Calculator is faster. For triangles without a right angle, use the Law of Sines or Law of Cosines calculators, and to bring a large or negative angle back into the range from 0° to 360°, the Coterminal Angle Calculator shows every step.

The three core ratios: SOH-CAH-TOA

In a right triangle, the three primary trig functions relate one acute angle to two of the three sides. Label the side opposite the angle, the side adjacent to it, and the hypotenuse (always the longest side, across from the 90° corner):

The mnemonic SOH-CAH-TOA packs all three into one phrase: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. Because the hypotenuse is the longest side, sine and cosine of an acute angle always fall between 0 and 1, while tangent can grow without bound as the angle approaches 90°.

The reciprocal ratios

Three more functions are simply the reciprocals of the first three. They appear constantly in calculus and physics identities, so the calculator reports them too:

Where a denominator hits zero (for example, cos 90° = 0 makes both tan and sec undefined), the calculator says undefined instead of showing a misleading huge number.

Degrees vs radians

The same physical angle can be written two ways. Degrees divide a circle into 360 parts; radians measure the angle by the arc it sweeps on a unit circle, so a full turn is 2π radians. Convert with:

That makes 180° = π ≈ 3.1416 radians, 90° = π/2 ≈ 1.5708, 60° = π/3 ≈ 1.0472, 45° = π/4 ≈ 0.7854 and 30° = π/6 ≈ 0.5236. Switching the unit at the top of the calculator converts the angle you typed, so 30 becomes π/6 and the ratios stay the same. In radian mode you can also type multiples of π directly, such as 2π/3 or 3pi/4.

How to use this trigonometry calculator

1. Pick a mode: "Trig ratios" for all six ratios of an angle, "Inverse" to find the angle for a ratio such as 0.5 or √3/2, or "Right triangle" to find missing sides.
2. Choose your unit: degrees or radians. In radians you can type π/6, 3π/4 or pi/4. The quick-pick chips load common angles for you.
3. Enter the angle (ratios mode), or the acute angle plus one side (triangle mode), choosing whether that side is the hypotenuse, the opposite, or the adjacent side.
4. Read the result: the large primary value updates as you type, with no submit button.
5. Scan the supporting cards: reciprocal ratios, the angle in degrees and radians with its coterminal and reference angles, and (in triangle mode) the step-by-step SOH-CAH-TOA derivation with a Pythagorean cross-check.

Worked example 1: ratios of 30°

Enter 30 in degrees. The calculator returns sin 30° = 0.5 exactly, cos 30° ≈ 0.8660 (that is √3/2), and tan 30° ≈ 0.5774 (that is 1/√3 = √3/3). The reciprocals come out as csc 30° = 2, sec 30° ≈ 1.1547 (2√3/3) and cot 30° ≈ 1.7321 (√3). Switching to Inverse mode and entering 0.5 returns arcsin 0.5 = 30°, confirming the round trip. In radians the same angle reads as π/6 ≈ 0.5236, and all the ratios are identical because the angle itself has not changed, only its label.

Worked example 2: a ladder against a wall

A ladder leans against a wall at a 70° angle to the ground and its foot is 1.2 m from the wall. The 1.2 m is the side adjacent to the 70° angle, so switch to triangle mode, set the angle to 70°, choose "Adjacent" (side b), and enter 1.2. These are the calculator's default triangle values. The solver returns the ladder length (the hypotenuse) as c = b ÷ cos 70° ≈ 3.51 m, the height it reaches as a = b × tan 70° ≈ 3.30 m, and the second angle as 20°. This is a textbook CAH/TOA problem, solved without you having to rearrange a single formula by hand.

Worked example 3: roof rise from run and angle

A roof slopes at 25° and has a horizontal run of 16 ft from the eave to the ridge line. The run is the adjacent side, so the vertical rise is a = 16 × tan 25° ≈ 7.46 ft, and the rafter length (hypotenuse) is 16 ÷ cos 25° ≈ 17.65 ft. The area of that right-triangle cross-section is about 59.7 ft². Carpenters use exactly this calculation, often phrased as "rise over run", to cut rafters to length. Since 12 × tan 25° ≈ 5.6, the same roof has a pitch of about 5.6/12, and the Roof Pitch Calculator converts between angles and pitches. If instead you already know the rise and run and only need the rafter (hypotenuse) length, the Pythagorean Theorem Calculator gets there without any angle at all.

Reference table: exact values of common angles

These angles show up so often that their values are worth memorizing. The exact forms use the square roots √2 ≈ 1.4142 and √3 ≈ 1.7321:

sin, cos and tan table for every 5° from 0° to 90°

Looking up a single value like cos 25° or cos 70°? This decimal table lists all three primary ratios (plus the radian measure) for every 5° across the first quadrant, computed to four places. Read across the row for your angle.

How to compute n × cos θ and n × sin θ

A search like "4 cos 30" or "20 cos 30" means "multiply the ratio by a number": evaluate the trig value first, then multiply. Below are worked answers to common multiples. Because cos 30° and sin 60° both equal √3/2 ≈ 0.866025, every row built on them is a multiple of that value; for example, 2 × cos 30° = √3 ≈ 1.7321.

Worked step by step: 4 × cos 30°

Order of operations matters: evaluate the trig function before multiplying. Here is "4 cos 30" broken into steps:

Who this calculator is for

• Students checking homework on the unit circle, trig ratios, or right-triangle problems.
• DIY builders and trades figuring out rafter lengths, ramp angles, stair stringers and ladder reach.
• Surveyors, drafters and makers converting between angles and distances on the fly.
• Programmers and game developers sanity-checking sin/cos values before wiring them into code (remembering that most languages use radians).
• Anyone who needs a quick degree-to-radian conversion or the exact value of a common angle.

Inverse trig functions explained

The inverse (or "arc") functions reverse the ratios: give arcsin a value and it returns the angle whose sine is that value. Because each ratio repeats endlessly around the circle, the inverses return only the principal value: arcsin lies from −90° to 90°, arccos from 0° to 180°, and arctan strictly between −90° and 90° (NIST DLMF §4.23). arcsin and arccos are defined only for values from −1 to 1, so arcsin 2 is undefined, while arctan accepts any real number. So arcsin(0.5) = 30°, arccos(0.5) = 60°, and arctan(1) = 45°. This is why entering an angle, reading its sine, and then taking arcsine may not return your starting angle if it was outside the principal range; the Inverse mode notes this under its results.

Key terms at a glance

• Hypotenuse: the side opposite the right angle; always the longest side of a right triangle.
• Opposite / adjacent: defined relative to the angle you are working with: the opposite side faces the angle, and the adjacent side touches it (other than the hypotenuse).
• Unit circle: a circle of radius 1 used to define sin and cos for any angle as the y- and x-coordinates of a point on the circle.
• Period: sin and cos repeat every 360° (2π); tan repeats every 180° (π). That periodicity is why large and negative angles are valid inputs.
• Principal value: the single angle an inverse function returns from its restricted output range.
• Reference angle: the acute angle between the terminal side and the x-axis, from 0° to 90°; the calculator shows it, along with the coterminal angle from 0° up to 360°, for every angle you enter.

Trig identities worth knowing

Beyond the raw ratios, a handful of identities tie the functions together and let you simplify expressions or check a result by a second route. The most fundamental is the Pythagorean identity, which falls straight out of the unit circle:

Two more families show up constantly. The co-function identities say that a function of an angle equals the co-function of its complement: sin θ = cos(90° − θ), tan θ = cot(90° − θ), and so on, which is exactly why the two acute angles of a right triangle trade their sines and cosines. The even/odd identities describe symmetry: cosine is even, so cos(−θ) = cos θ, while sine and tangent are odd, so sin(−θ) = −sin θ and tan(−θ) = −tan θ. These are the reason a negative angle on this calculator flips the sign of the sine and tangent but leaves the cosine alone. If you ever doubt a value, plug it back into sin² θ + cos² θ and confirm it lands on 1.

The unit circle and the four quadrants

The cleanest way to extend trig past the 0° to 90° of a right triangle is the unit circle: a circle of radius 1 centered on the origin. For any angle measured counter-clockwise from the positive x-axis, the point where the angle's ray meets the circle has coordinates (cos θ, sin θ). Tangent is then the slope of that ray, sin θ ÷ cos θ. This single picture explains every sign rule you need:

• Quadrant I (0° to 90°): x and y are both positive, so all six functions are positive.
• Quadrant II (90° to 180°): x is negative and y positive, so only sine (and its reciprocal cosecant) stays positive.
• Quadrant III (180° to 270°): both coordinates are negative, so only tangent (and cotangent) is positive.
• Quadrant IV (270° to 360°): x is positive and y negative, so only cosine (and secant) is positive.

The classic mnemonic "All Students Take Calculus" names which function is positive in each quadrant: All, Sine, Tangent, Cosine. Because the circle repeats, adding or subtracting a full turn (360° or 2π) lands you on the same point. That periodicity is what lets this calculator accept an angle like 750° and treat it as 30°.

Related concepts and where to go next

Trigonometry connects to several other tools on this site. To find a missing side of a right triangle when you already know two sides (no angle needed), the Pythagorean Theorem Calculator applies a² + b² = c² directly. For triangles without a right angle, the Law of Sines and Law of Cosines calculators take over, and the Triangle Calculator handles general combinations of sides and angles. The Arc Length Calculator uses radians to turn an angle into a distance along a circle, and the Coterminal Angle Calculator reduces any angle to the range from 0° to 360°. For roots like √2 and √3 that appear in exact trig values, the Square Root Calculator helps, and for broader number crunching with built-in trig keys, try the Scientific Calculator.

Sources

• NIST Digital Library of Mathematical Functions: §4.14 Definitions and Periodicity (sin, cos, tan and the reciprocal functions).
• NIST Digital Library of Mathematical Functions: §4.23 Inverse Trigonometric Functions (principal values of arcsin, arccos and arctan).
• Common Core State Standards for Mathematics: High School Functions, Trigonometric Functions (HSF-TF) (radians, the unit circle, exact values for π/6, π/4 and π/3, inverse functions, sin² θ + cos² θ = 1).
• Common Core State Standards for Mathematics: High School Geometry, Similarity, Right Triangles and Trigonometry (HSG-SRT) (trig ratios in right triangles and solving right triangles).