The six trigonometric functions — sine, cosine, tangent and their reciprocals cosecant, secant and cotangent — turn an angle into a ratio. This calculator evaluates all six at once for any angle, positive or negative, small or many turns around the circle. When the angle is a special one, it gives the exact radical form (such as √3/2) next to the decimal, and the steps show how the quadrant and reference angle produce each sign.
How to use the trigonometric functions calculator
- Type the angle. Plain numbers work (150, −45, 765), and so do expressions such as
2pi/3orsqrt(2). - Choose the unit: degrees, radians or gradians (400 gradians make a full turn).
- Pick a headline function if you want one value shown large. All six are always listed on the tape.
- Read the results: exact and decimal values, the angle converted to the other unit, the coterminal angle between 0° and 360°, the quadrant and the reference angle. The unit-circle sketch marks the angle and its reference angle, and Show the work walks through each step.
Trig function formulas
Place the angle θ in standard position, with its vertex at the origin and its initial side on the positive x-axis. If the terminal side meets the unit circle at the point (x, y), then:
The other three are reciprocals:
To convert between units, use 180° = π radians = 200 gradians:
Worked example: θ = 840°
1. Find a coterminal angle. 840° is more than two full turns. Subtract 2 × 360°: 840° − 720° = 120°.
2. Locate the quadrant. 120° lies between 90° and 180°, so it is in Quadrant II, where only sine and cosecant are positive.
3. Find the reference angle. 180° − 120° = 60°.
4. Apply the signs. sin 60° = √3/2 and cos 60° = 1/2, so sin 840° = √3/2 ≈ 0.8660254038 and cos 840° = −1/2.
5. Build the rest. tan 840° = (√3/2) ÷ (−1/2) = −√3. The reciprocals are csc 840° = 2√3/3, sec 840° = −2 and cot 840° = −√3/3.
In radians the same angle is 14π/3 ≈ 14.6607657168, and entering 14pi/3 with Radians selected gives identical values.
Reference angles and the ASTC rule
Every angle shares its trig values, up to sign, with an acute reference angle. The quadrant decides the sign. A common memory aid is ASTC — “All Students Take Calculus” — read counterclockwise from Quadrant I:
| Quadrant | Angle range | Reference angle | Positive functions |
|---|---|---|---|
| I | 0° to 90° | θ | all six |
| II | 90° to 180° | 180° − θ | sin, csc |
| III | 180° to 270° | θ − 180° | tan, cot |
| IV | 270° to 360° | 360° − θ | cos, sec |
Angles that land exactly on an axis (0°, 90°, 180°, 270°) belong to no quadrant. There, one coordinate is zero, which is why some functions are undefined.
When a trig function is undefined
Tangent and secant divide by cos θ, so they are undefined whenever cos θ = 0: at 90°, 270° and every angle 90° + 180°k. Cotangent and cosecant divide by sin θ, so they are undefined at 0°, 180° and every multiple of 180°. A floating-point calculator usually prints something like 1.633 × 10¹⁶ for tan 90° because π cannot be stored exactly. This tool recognizes the exact angle and reports Undefined instead, and it shows sin 180° as exactly 0 rather than a tiny leftover such as 1.2 × 10⁻¹⁶.
Exact values beyond 30°, 45° and 60°
Most textbooks stop at the 30-60-90 and 45-45-90 triangles, but several other angles have exact square-root forms:
| Angle | sin | cos | tan |
|---|---|---|---|
| 15° | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 |
| 18° | (√5 − 1)/4 | √(10 + 2√5)/4 | √(25 − 10√5)/5 |
| 22.5° | √(2 − √2)/2 | √(2 + √2)/2 | √2 − 1 |
| 36° | √(10 − 2√5)/4 | (1 + √5)/4 | √(5 − 2√5) |
The 18° and 36° values come from the regular pentagon and involve the golden ratio. For angles such as 20° or 37°, no short real-radical form exists, so the calculator shows decimals rounded to 10 places.
Degrees, radians and calculator mode
The most common trig mistake is a calculator in the wrong mode. Typing sin 30 in radian mode returns −0.9880316241, the sine of 30 radians. If you need a quick check, remember that sin 30° must equal exactly 0.5. To convert many angles at once, use the angle converter; for angles written as fractions of π, the trig functions of π calculator keeps everything exact.
Frequently asked questions
Why does my calculator say sin 30 = −0.988?
It is in radian mode, so it evaluated the sine of 30 radians (about 1,718.87°), not 30 degrees. Switch the mode to degrees, or here choose Degrees as the unit, and sin 30° = 1/2.
Why is tan 90° undefined?
Tangent is sin θ ÷ cos θ, and cos 90° = 0. Division by zero has no value, so tan 90° (and sec 90°) is undefined. Near 90° the tangent grows without bound: tan 89.9999° is about 572,958.
What is a reference angle?
It is the acute angle between the terminal side of your angle and the x-axis, always between 0° and 90°. Every trig value of the angle equals the value at the reference angle, give or take a sign set by the quadrant.
Which angles have exact values?
Every multiple of 30° or 45° gives the familiar forms with √2 and √3. This calculator also returns exact radicals for multiples of 15°, 18° and 22.5°, such as sin 15° = (√6 − √2)/4 and cos 36° = (1 + √5)/4.
Can I type an expression instead of a number?
Yes. The angle box accepts expressions such as 2pi/3, 180/7 or 45 + 30. With Radians selected, pi/6 is read as π/6 radians; with Degrees selected, pi means about 3.14159 degrees, and a note reminds you.