Exact Value of Trig Functions Calculator
Evaluate sine, cosine, tangent, cotangent, secant, and cosecant for one angle, with exact radical forms shown for common special angles and precise decimal values for any valid angle.
Angle input
Live results
Function values
| Function | Exact form | Decimal value |
|---|---|---|
| Sine, sin(θ) | 1/2 | 0.5 |
| Cosine, cos(θ) | √3/2 | 0.8660254038 |
| Tangent, tan(θ) | √3/3 | 0.5773502692 |
| Cotangent, cot(θ) | √3 | 1.7320508076 |
| Secant, sec(θ) | 2√3/3 | 1.1547005384 |
| Cosecant, csc(θ) | 2 | 2 |
How to use this exact value of trig functions calculator
What this calculator does
This calculator evaluates all six standard trigonometric functions for one angle: sine, cosine, tangent, cotangent, secant, and cosecant. It reports a decimal value for every defined function and, when the angle is one of the supported special-angle families, also gives an exact radical form. It is useful for checking unit-circle work, verifying right-triangle exercises, preparing exact values for algebraic manipulations, and comparing a symbolic result with its decimal approximation. It does not derive a closed-form radical expression for every possible angle; many angles have exact algebraic descriptions that are too complicated to be useful in elementary work.
When to use it
Use it when you need to check a unit-circle table, confirm the sign of a trig function in a particular quadrant, convert a familiar degree angle to radians before evaluating it, or verify whether tangent, cotangent, secant, or cosecant is undefined at an axis angle. The underlying unit-circle definition and reference-angle method are covered in the OpenStax unit-circle section.
How to calculate
- The calculator opens with a complete 30° demonstration. Its exact and decimal results are already visible, and the example XLSX workbook is immediately available from Download Excel.
- Replace Angle with your value. In degrees, enter an ordinary decimal such as 225 or 22.5. In radians, you may enter a decimal such as 0.7853981634 or a simple π expression such as π/4 or 3π/4.
- Choose Angle unit. Switching from degrees to radians or back converts the current valid value rather than changing only the label. Read the live cards or the function table, then export the same canonical values with Download Excel. Reset clears the demonstration angle and results; Excel export is then disabled until a complete valid angle is entered again.
Input guide
Angle is required. It accepts a finite real angle with no fixed magnitude limit, because trigonometric functions are periodic. Degree mode uses a period of 360°, while radian mode uses a period of 2π. Ordinary decimal notation is accepted in both modes. Scientific notation and ambiguous grouped-number formats are rejected rather than silently reinterpreted. In radian mode, simple π notation is also accepted. For example, 30 in degree mode produces the startup values, while π/6 in radian mode represents the same angle. Increasing or decreasing the angle changes the terminal point on the unit circle; because the functions are periodic, a larger numeric angle does not imply a larger trig value.
Angle unit is required and has two options: Degrees (°) and Radians (rad). Use degrees for familiar geometric angles such as 30°, 45°, or 60°. Use radians for calculus and analytic work, where π/6, π/4, and π/3 are common. A frequent mistake is entering a degree number while radians are selected; for example, 30 radians is not the same angle as 30 degrees.
Output guide
Normalized reduces the angle to one revolution, which makes periodic equivalence easier to see. Reference angle gives the acute or axis angle used to transfer known special-angle values into the correct quadrant. Quadrant identifies the sign pattern, while Exact form available tells you whether this calculator recognized a supported special angle. The six function outputs are dimensionless identities: Sine and Cosine are the unit-circle y- and x-coordinates; Tangent is sin(θ)/cos(θ); Cotangent is cos(θ)/sin(θ); Secant is 1/cos(θ); and Cosecant is 1/sin(θ). Undefined appears when the relevant denominator is zero. The reciprocal definitions are explained in the OpenStax guide to the other trigonometric functions.
Worked example
With the startup input Angle = 30 and Angle unit = Degrees (°), the unit-circle coordinates are cos(30°) = √3/2 and sin(30°) = 1/2. Therefore tan(30°) = (1/2)/(√3/2) = √3/3. Taking reciprocals gives cot(30°) = √3, sec(30°) = 2√3/3, and csc(30°) = 2. The displayed decimals are 0.5, 0.8660254038, 0.5773502692, 1.7320508076, 1.1547005384, and 2 respectively. These are the same values placed into the startup workbook.
How exact trig values are obtained
For special angles, exact values come from a small collection of geometric constructions and identities. The 45° value follows from a 45-45-90 triangle, while 30° and 60° come from a 30-60-90 triangle. Half-angle formulas produce useful values such as 15° and 22.5°, and symmetry across the axes supplies corresponding values in other quadrants. This calculator recognizes common special-angle families including multiples derived from 15°, 18°, and 22.5° reference angles, then applies the correct quadrant signs.
tan(θ) = sin(θ) / cos(θ)
sec(θ) = 1 / cos(θ), csc(θ) = 1 / sin(θ), cot(θ) = cos(θ) / sin(θ)
Periodicity matters just as much as the base values. Sine and cosine repeat after 2π radians, and tangent repeats after π radians. The NIST Digital Library of Mathematical Functions gives formal definitions and periodicity properties for the trigonometric functions. In practical unit-circle work, reducing a large positive or negative angle to one revolution is usually the cleanest first step.
Interpreting exact and decimal forms
An exact form preserves the mathematical identity without rounding. For example, √3/2 is preferable to 0.8660254038 when you will square the value, combine it with another radical, or prove an identity. A decimal is useful for measurement, plotting, or numerical comparison. When the calculator says that no simple supported exact form is available, the decimal result is still computed from the angle directly. At axis angles, do not replace an undefined reciprocal or quotient with a very large number: the function is undefined because its denominator is exactly zero. That distinction is important when solving equations and analyzing limits.