📐 Trigonometry Calculator
By ToolNimba Math Team · Updated 2026-06-19
Enter an angle to see all six trigonometric functions.
This trigonometry calculator gives you all six trigonometric functions of an angle in one place. Enter an angle in degrees or radians and read off sin, cos and tan along with their reciprocals csc, sec and cot. Switch to inverse mode to go the other way: type a ratio and the calculator returns the angle using arcsin, arccos or arctan. Cases that are mathematically undefined, such as tan of 90 degrees, are clearly flagged rather than shown as a misleading number.
What is the Trigonometry Calculator?
Trigonometry connects the angles of a triangle to the ratios of its sides. The three core functions are defined on a right triangle: sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, and tangent is the opposite over the adjacent. The same idea extends to any angle using the unit circle, where a point at angle theta has coordinates (cos theta, sin theta), and tangent is the slope sin theta over cos theta. That is why these functions repeat every full turn and take values between -1 and 1 for sine and cosine.
Each core function has a reciprocal. Cosecant (csc) is 1 over sine, secant (sec) is 1 over cosine, and cotangent (cot) is 1 over tangent (equivalently cosine over sine). Because you cannot divide by zero, a reciprocal is undefined wherever its base function equals zero. So csc and cot are undefined at 0 and 180 degrees (where sine is zero), and sec and tan are undefined at 90 and 270 degrees (where cosine is zero). A good calculator reports undefined in these spots instead of returning a huge rounded number caused by floating-point error.
The inverse functions answer the reverse question: given a ratio, what angle produces it? Since the trig functions repeat, the inverses return a single principal value. Arcsin returns an angle from -90 to 90 degrees, arccos returns 0 to 180 degrees, and arctan returns -90 to 90 degrees. Arcsin and arccos only accept inputs between -1 and 1, because sine and cosine never go outside that range. Arctan accepts any real number, since tangent runs from minus infinity to plus infinity.
When to use it
- Checking homework answers for sin, cos and tan of common angles like 30, 45 and 60 degrees.
- Finding an unknown angle in a right triangle from a known side ratio using inverse trig.
- Converting between an angle and its trig ratios when working in either degrees or radians.
- Confirming where functions like tan, sec, csc and cot are undefined before sketching a graph.
How to use the Trigonometry Calculator
- Pick a mode: angle to functions, or inverse (ratio to angle).
- Choose whether you are working in degrees or radians.
- In function mode, type the angle to see sin, cos, tan, csc, sec and cot at once.
- In inverse mode, choose arcsin, arccos or arctan and enter the ratio to get the angle.
Formula & method
Worked examples
Find all six trig functions of 30 degrees.
- Convert: 30 degrees = 30 x pi ÷ 180 = 0.5235988 radians
- sin 30 = 0.5
- cos 30 = 0.8660254
- tan 30 = sin ÷ cos = 0.5 ÷ 0.8660254 = 0.5773503
- csc 30 = 1 ÷ 0.5 = 2
- sec 30 = 1 ÷ 0.8660254 = 1.1547005
- cot 30 = 1 ÷ 0.5773503 = 1.7320508
Result: sin 0.5, cos 0.8660254, tan 0.5773503, csc 2, sec 1.1547005, cot 1.7320508
Use inverse mode: which angle has a sine of 0.5? (arcsin 0.5)
- arcsin gives the principal angle whose sine is 0.5
- arcsin(0.5) = 0.5235988 radians
- Convert to degrees: 0.5235988 x 180 ÷ pi = 30 degrees
- Note the principal range of arcsin is -90 to 90 degrees
Result: arcsin(0.5) = 30 degrees = 0.5235988 radians
Trig functions at common angles (undefined where a division by zero occurs)
| Angle | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | undefined | 1 | undefined |
| 30 | 0.5 | 0.8660 | 0.5774 | 2 | 1.1547 | 1.7321 |
| 45 | 0.7071 | 0.7071 | 1 | 1.4142 | 1.4142 | 1 |
| 60 | 0.8660 | 0.5 | 1.7321 | 1.1547 | 2 | 0.5774 |
| 90 | 1 | 0 | undefined | 1 | undefined | 0 |
| 180 | 0 | -1 | 0 | undefined | -1 | undefined |
Inverse trig functions: allowed input and output range
| Function | Accepts input | Returns angle in |
|---|---|---|
| arcsin | -1 to 1 | -90 to 90 degrees |
| arccos | -1 to 1 | 0 to 180 degrees |
| arctan | any real number | -90 to 90 degrees |
Common mistakes to avoid
- Mixing up degrees and radians. A calculator set to radians will give a very different answer for the same number than one set to degrees. sin 30 in degree mode is 0.5, but sin 30 in radian mode is about -0.988. Always confirm the unit toggle matches the angle you typed.
- Expecting a number where the function is undefined. tan 90 degrees and sec 90 degrees are undefined because cosine is zero there, and csc 0 and cot 0 are undefined because sine is zero. Some calculators show a huge value instead, which is just floating-point error, not a real result.
- Feeding arcsin or arccos a value outside -1 to 1. Sine and cosine never leave the range -1 to 1, so there is no real angle whose sine is, say, 2. Entering such a value into arcsin or arccos has no answer. Only arctan accepts numbers of any size.
- Forgetting inverse functions return only one angle. Because trig functions repeat, many angles share the same ratio. The inverse returns just the principal value in a fixed range. To find every solution you add multiples of the period or reflect into other quadrants.
Glossary
- Sine (sin)
- The ratio of the side opposite an angle to the hypotenuse in a right triangle; the y coordinate on the unit circle.
- Cosine (cos)
- The ratio of the side adjacent to an angle to the hypotenuse; the x coordinate on the unit circle.
- Tangent (tan)
- Sine divided by cosine, equal to the opposite side over the adjacent side; undefined where cosine is zero.
- Reciprocal functions
- csc, sec and cot, equal to 1 over sin, cos and tan respectively; undefined wherever the base function is zero.
- Radian
- An angle unit where a full turn is 2 pi radians; one radian is about 57.2958 degrees.
- Principal value
- The single angle an inverse trig function returns from its fixed output range, rather than all possible solutions.
Frequently asked questions
How do I calculate sin, cos and tan of an angle?
Choose your unit (degrees or radians), type the angle, and the calculator returns sin, cos and tan together with the reciprocals csc, sec and cot. Internally it converts degrees to radians, then applies the standard sine and cosine functions and forms the ratios from them.
Why is tan 90 degrees undefined?
Tangent is sine divided by cosine. At 90 degrees cosine is zero, so tan 90 would require dividing by zero, which has no value. The same happens for sec 90. Likewise csc and cot are undefined at 0 and 180 degrees, where sine is zero.
What is the difference between degrees and radians?
Both measure angles. A full circle is 360 degrees or 2 pi radians, so 180 degrees equals pi radians. To convert, multiply degrees by pi over 180 to get radians, or multiply radians by 180 over pi to get degrees. This calculator lets you switch between the two with one toggle.
How do inverse trig functions work?
Inverse functions take a ratio and return the angle that produces it. Use the inverse mode, pick arcsin, arccos or arctan, and enter the ratio. The result is the principal value: arcsin and arctan return -90 to 90 degrees, while arccos returns 0 to 180 degrees.
Why does arcsin reject numbers above 1?
Sine of any angle always falls between -1 and 1, so there is no real angle whose sine is greater than 1 or less than -1. Arcsin and arccos therefore only accept inputs in that range. Arctan has no such limit because tangent can be any real number.
What are csc, sec and cot?
They are the reciprocal trig functions: cosecant (csc) is 1 over sine, secant (sec) is 1 over cosine, and cotangent (cot) is 1 over tangent, equal to cosine over sine. Each is undefined wherever the function it is built from equals zero.