Trig Function Evaluator
Pick a trigonometric function and an angle. The evaluator converts degrees to radians when needed, then computes the value. The reciprocal functions csc, sec and cot are derived from sin, cos and tan, and undefined cases are reported clearly.
Six functions, three computations
There are six trigonometric functions, but only three of them are independent. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent, and everything about their behaviour — including where they blow up — follows from that.
This evaluator computes any of the six at an angle given in degrees or radians, showing the conversion when one is needed and reporting the undefined cases rather than returning an enormous number.
The reciprocal three are derived rather than implemented separately, which is why they are undefined exactly where their partners are zero. Cosecant fails wherever sine vanishes, secant wherever cosine does, and cotangent wherever tangent does.
The unit selector matters more here than almost anywhere else. Both units accept the same numbers without complaint, and an angle of 30 means something entirely different in each — so the conversion line, present only in degree mode, is the confirmation worth reading.
How to use this calculator
- Choose the function Any of the six. The three reciprocals are computed from their partners rather than independently.
- Enter the angle Any real number, including negatives and values beyond a full turn. Nothing needs reducing first.
- Set the unit Degrees or radians. This is the one input that can silently produce a completely wrong answer, since both accept identical values.
- Read the conversion line if working in degrees It shows the radian equivalent before evaluation. Its absence means radians were assumed.
The formula, and where it comes from
csc θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ θ_rad = θ_deg × π/180
Sine and cosine are the primary pair, and everything else is built from them. Tangent is their ratio, and the three reciprocals invert the first three — so six functions rest on two underlying quantities.
That layering explains the undefined points. Tangent is undefined where cosine is zero because the division fails; cosecant is undefined where sine is zero for the same reason one step further out. None of these are special cases in the code, only divisions that cannot be performed.
The degree conversion happens first when it is needed, and it is genuine rather than cosmetic — the underlying functions work in radians, so multiplying by π and dividing by 180 is a required step rather than a display choice.
A result that is not finite, or whose magnitude exceeds a very large threshold, is reported as undefined rather than shown. That threshold is what catches the cases where floating-point arithmetic gives a huge number instead of the infinity the mathematics calls for.
What each input means
- func Function — form field “Function”
- One of the six. The reciprocal three inherit their undefined points from the function they invert.
- θ Angle — form field “Angle”
- Any real number. Adding a full turn leaves sine, cosine and their reciprocals unchanged; tangent and cotangent repeat twice as often.
- unit Unit — form field “Unit”
- Degrees or radians. It determines what the angle number means and nothing else about the calculation.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A standard angle
Sine of thirty degrees — one of the handful of angles whose exact value is worth knowing by heart.
Inputs Function = sin, Angle = 30, Unit = deg
- Convert to radians 30° × π/180 = 0.523599 rad
- sin sin(30°) = 0.5
Result sin(30°) = 0.5
The answer is exactly one half, and the conversion line shows the angle first becoming roughly 0.5236 radians. The tidy result comes from the mathematics, not from the intermediate value, which is irrational.
Thirty degrees is one of the angles a right triangle with sides in the ratio 1, √3, 2 produces. Those exact values are what make the standard angles worth memorising rather than looking up.
A complementary angle
Cosine of sixty degrees. Sixty and thirty are complementary, which is why this answer matches the previous one.
Inputs Function = cos, Angle = 60, Unit = deg
- Convert to radians 60° × π/180 = 1.0472 rad
- cos cos(60°) = 0.5
Result cos(60°) = 0.5
The result is also one half. Cosine of an angle equals sine of its complement, and thirty and sixty sum to a right angle — which is the relationship the co- prefix in cosine refers to.
That symmetry means the standard angle values need only be learned once. Every cosine value is a sine value read from the other end of the range.
An angle already in radians
Tangent of a quarter of π, entered as a decimal with the unit set to radians.
Inputs Function = tan, Angle = 0.7853981634, Unit = rad
- Angle 0.785398 rad
- tan tan(0.785398 rad) = 1
Result tan(0.785398) = 1
No conversion line appears, which confirms the unit selector was set to radians. The answer is 1, since a quarter turn of π corresponds to forty-five degrees where sine and cosine are equal.
Entering the same number in degree mode would give a value near 0.0137 — the tangent of less than one degree. Both are valid inputs and only the selector distinguishes them.
Reading the result
Where the functions are undefined
Tangent and secant fail at ninety degrees and every half-turn from it, where cosine is zero. Cotangent and cosecant fail at zero and every half-turn from there, where sine is zero. The tool names these rather than returning a huge number.
Ranges differ between the families
Sine and cosine never leave the interval from −1 to 1. Tangent and cotangent take every real value. Cosecant and secant, being reciprocals of bounded functions, never fall strictly between −1 and 1 — a result inside that range is a sign the wrong function was selected.
Everything repeats
Sine, cosine and their reciprocals repeat every full turn; tangent and cotangent every half turn. An angle need not be reduced before entry, since the periodicity is handled by the underlying functions.
When you would use this
Evaluating a term in a larger calculation
Any formula containing a trigonometric term needs its value at a specific angle, and the reciprocal functions in particular are rarely available directly on a calculator.
Checking a remembered exact value
The standard angles have exact values that are easy to misremember. Evaluating one confirms it in a second, and the decimal makes clear which of two similar-looking surds is correct.
Assumptions and limitations
What this calculator assumes
- Angles are measured from the positive horizontal axis in the usual anticlockwise sense.
- Degrees are converted to radians before evaluation.
- The reciprocal functions are computed from their partners, so they inherit their undefined points.
- A non-finite or extremely large result is reported as undefined rather than displayed.
Where it stops being the right tool
- One function and one angle at a time.
- No inverse functions — recovering an angle from a value is the trig equation solver's job.
- Results are decimals, so a value of one half over the square root of two is shown rounded rather than as a surd.
- Gradians are not offered as a unit.
Common mistakes
Leaving the unit on the wrong setting
Why it happens. Both units accept the same numbers, and 30 is a perfectly plausible angle in either. The results differ completely and neither looks obviously wrong.
How to avoid it. Look for the conversion line. It appears only in degree mode, so its presence or absence identifies which unit was applied before the answer is read.
Confusing cosine with cosecant
Why it happens. The names are similar and both begin with the same three letters, which makes the wrong one easy to select from a list.
How to avoid it. Cosecant is the reciprocal of sine, not of cosine — the naming is genuinely counterintuitive. Secant is the one paired with cosine.
Expecting a number where the function is undefined
Why it happens. Every other angle returns a value, so a message in its place reads as a failure rather than as a fact about the function.
How to avoid it. Check whether the underlying function is zero there. Tangent at ninety degrees is undefined because cosine vanishes, not because the calculation failed.
Key terms
Frequently asked questions
Can I work in degrees?
Yes — choose the unit and degree inputs are converted to radians before evaluation, with the conversion shown as its own step. That line is also the quickest way to confirm the selector was set as you intended.
Why is tangent undefined at 90 degrees?
Because tangent is sine divided by cosine, and cosine is zero there. The division cannot be performed, so the tool reports it as undefined rather than returning the very large number floating-point arithmetic would otherwise produce.
What are csc, sec and cot?
The reciprocals of sine, cosine and tangent respectively. Note the pairing is not what the names suggest: cosecant inverts sine, and secant inverts cosine.
Do I need to reduce a large angle first?
No. The functions are periodic and handle any real input, so 400 degrees and 40 degrees give the same sine. Negative angles are equally acceptable and simply measure clockwise.