Trigonometry Calculator
Enter an angle and this returns all six trigonometric ratios — sin, cos, tan, csc, sec and cot — along with the quadrant it lands in, its reference angle, and its coordinates on the unit circle. Switch the mode to go the other way: give a sine, cosine or tangent and get the angle back from arcsin, arccos or arctan. Degrees, radians and gradians are interchangeable throughout, and every answer is shown in all three.
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sin θ
0.500000
cos θ
0.866025
Details
Updates as you typeThe inverse modes read the ratio box instead of the angle box, then report all six ratios for the angle they found.
A full turn is 360°, 2π rad, or 400 grad. Switching converts the angle already typed rather than reinterpreting it.
One full turn either way. Anything outside it is reduced to a coterminal angle, so 400° is answered as 40°.
Read by the inverse modes only. Sine and cosine live in −1 to 1; a tangent has no limit.
Summary
sin θ
0.500000
- sin θ
- 0.500000
- cos θ
- 0.866025
- tan θ
- 0.577350
- csc θ
- 2.000000
- sec θ
- 1.154701
- cot θ
- 1.732051
- Every value here is read off the unit circle: sin θ is the y-coordinate of the point θ from the positive x-axis, cos θ is the x-coordinate, and tan θ is y ÷ x. On an acute angle that is exactly SOH-CAH-TOA, with the hypotenuse scaled to 1.
How this is calculated
- Angle θ (given)
- 30.0000 °
- θ in degrees
- 30.0000 °
- θ in radians
- 0.523599 rad
- θ in gradians
- 33.3333 grad
- Quadrant
- I — sin, cos and tan all positive
- Coterminal angle in one turn
- 30.0000 °
- Reference angle
- 30.0000 °
- Unit circle x = cos θ
- 0.866025
- Unit circle y = sin θ
- 0.500000
Angle (°)
Compare scenarios
See how one change moves the result
- CurrentYour inputs as they stand0.500000Current
- Angle θ° 380.615661
Frequently asked questions
What are the six trigonometric functions?
Three primary and three reciprocal. On a unit circle, the point at angle θ from the positive x-axis has coordinates (cos θ, sin θ): sine is the y-coordinate, cosine is the x-coordinate, and tangent is y ÷ x. The other three are their reciprocals — cosecant is 1 ÷ sin θ, secant is 1 ÷ cos θ, and cotangent is cos θ ÷ sin θ. For an acute angle in a right triangle the same three read as opposite ÷ hypotenuse, adjacent ÷ hypotenuse and opposite ÷ adjacent, which is SOH-CAH-TOA.
Why is tan 90° undefined instead of infinite?
At 90° the point on the unit circle is (0, 1), so tangent asks for 1 ÷ 0. That has no value — not a very large one. Approach 90° from below and the tangent grows without limit; approach from above and it falls without limit, so there is no single number the two sides agree on. Many calculators print something like 1.6e16 there, which is not a tangent at all but the rounding error in their own value of π ÷ 2. This one says undefined, and names which coordinate went to zero.
What is a reference angle, and why does it matter?
It is the acute angle between the terminal side and the x-axis, always between 0° and 90°. Every trigonometric ratio of an angle equals the ratio of its reference angle, up to a sign — so sin 150° is sin 30° and cos 210° is −cos 30°. That is what lets a single table of values from 0° to 90° serve the whole circle, and it is why the quadrant is reported alongside: the quadrant supplies the sign, and the reference angle supplies the number.
Should I use degrees or radians?
Degrees for geometry, surveying and navigation; radians for calculus and anything involving rates of change, because the derivative of sin x is cos x only when x is in radians. Gradians survive in some European surveying, where a right angle is a round 100. A full turn is 360°, 2π radians, or 400 gradians, so one degree is π ÷ 180 radians and 10 ÷ 9 gradians. This calculator reports every answer in all three, so a mismatch between your calculator and your textbook is visible rather than silent.
Why does arcsin only give one answer?
Because infinitely many angles share a sine. Inverse functions return the principal value — one angle from an agreed window — or they would not be functions at all. arcsin and arctan answer in −90° to 90°, and arccos in 0° to 180°. To recover the others within one turn: for a sine, the second solution is 180° − θ; for a cosine it is −θ; for a tangent it is θ + 180°. Then add any whole number of turns.
Why does the calculator refuse a sine of 2?
Because no angle has one. Sine and cosine are coordinates of a point on a circle of radius 1, so both stay between −1 and 1 for every real angle — which is why the ratio field narrows to that range in arcsin and arccos mode. Tangent has no such limit, so arctan accepts anything. A sine above 1 does have a meaning for complex angles, but not for the triangles and rotations this calculator is about, so it is refused against the field rather than answered with a NaN.