Trigonometry Calculator

Type an angle like 30° and this trigonometry calculator instantly returns sin 30° = 0.5, cos 30° ≈ 0.8660 and tan 30° ≈ 0.5774, along with the three reciprocal ratios and the inverse functions. Switch to triangle mode and the same angle plus one side length solves the whole right triangle — the other two sides, the second acute angle, the area and the perimeter. Whether you are checking trig homework, sizing a roof pitch, or working out how long a ladder needs to be, this tool turns angles into numbers (and numbers back into angles) in one place.

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 shows a dash instead of 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. Toggle the unit at the top of the calculator and every result re-expresses itself automatically — this is the bidirectional, instant-update behavior you would expect from a good converter.

How to use this trigonometry calculator

1. Pick a mode: "Trig ratios" for sin/cos/tan of an angle, or "Solve a right triangle" to find missing sides.
2. Choose your unit: degrees or radians. 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 instantly: the large primary value updates as you type — no submit button needed.
5. Scan the supporting cards: reciprocal ratios, the inverse-function values, 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). The reciprocals come out as csc 30° = 2, sec 30° ≈ 1.1547 and cot 30° ≈ 1.7321. Feeding 0.5 back into arcsine returns 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 (b)", and enter 1.2. 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 pitch

A roof has a 25° pitch and 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.

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:

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 and arctan lie in [-90°, 90°], and arccos in [0°, 180°]. 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 exact starting angle if it was outside the principal range — the calculator notes this where it matters.

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, 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.

Related concepts and where to go next

Trigonometry connects to several other tools. To raise numbers to powers inside an identity, use the Exponent Calculator; for roots like √2 and √3 that appear in exact trig values, the Square Root Calculator helps. For broader number crunching with built-in trig keys, try the Scientific Calculator. And for the percentage and averaging steps that often surround a trig problem, the Percentage and Average calculators are one click away below.