Inverse Trig Calculator

Find principal angles, domains, and reference triangle values. Plot function behavior with downloadable result reports. Solve inverse trig tasks faster with clean, guided calculations.

Calculator Form

Input Mode

Inverse Function

Output Unit

Ratio Value

Use this for direct entries like sin ratio, tan ratio, sec ratio, or cot ratio.

Decimal Precision

Choose between 2 and 12 decimal places.

Example Data Table

Inverse Function	Input	Angle (Degrees)	Angle (Radians)	Note
arcsin	0.5	30	0.523599	Common acute sine value.
arccos	0	90	1.570796	Midpoint of the cosine range.
arctan	1	45	0.785398	Equal opposite and adjacent lengths.
arcsec	2	60	1.047198	Because sec(60°) equals 2.
arccsc	2	30	0.523599	Because csc(30°) equals 2.
arccot	1	45	0.785398	Principal acute cotangent angle.

Formula Used

Core inverse relationships
1. θ = sin⁻¹(x), valid when -1 ≤ x ≤ 1.
2. θ = cos⁻¹(x), valid when -1 ≤ x ≤ 1.
3. θ = tan⁻¹(x), valid for all real x.
4. θ = sec⁻¹(x) = cos⁻¹(1/x), valid when |x| ≥ 1.
5. θ = csc⁻¹(x) = sin⁻¹(1/x), valid when |x| ≥ 1.
6. θ = cot⁻¹(x) is computed using an arctangent adjustment so the principal angle stays inside (0, π).

Side-based ratio formulas
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
sec(θ) = hypotenuse / adjacent
csc(θ) = hypotenuse / opposite
cot(θ) = adjacent / opposite

Principal-value ranges
Inverse functions return one principal angle, not every coterminal angle.
This keeps outputs consistent for engineering, algebra, geometry, and graphing workflows.

How to Use This Calculator

Select the input mode: direct ratio entry or side-based ratio.
Choose the inverse trig function you need.
Enter either the ratio itself or the matching side pair.
Pick degrees or radians for the displayed answer.
Set your desired decimal precision.
Press Calculate Inverse Trig.
Review the solved angle, range, verification value, companion trig values, and graph.
Use the CSV or PDF buttons to export the result summary.

FAQs

1. Why does arcsin reject numbers greater than 1?

Sine values from real angles stay between -1 and 1. Any larger magnitude would require a non-real angle, so the calculator blocks it.

2. Why does arccos return only one answer?

Inverse cosine uses a principal range from 0 to π. Many angles can share one cosine value, but the inverse function returns the standard principal angle only.

3. What is the difference between ratio mode and side mode?

Ratio mode lets you enter a prepared trig value directly. Side mode calculates that value from the appropriate side pair before solving the inverse angle.

4. Why are secant and cosecant restricted to |x| ≥ 1?

Secant and cosecant are reciprocals of cosine and sine. Since sine and cosine stay within -1 to 1, their reciprocals must have magnitude at least 1.

5. Why can arctan accept every real number?

Tangent covers all real outputs over its principal interval. Because of that, arctan can map any real input back to one principal angle.

6. What does the verification value show?

It applies the matching forward trig function to the solved angle. The result should closely match your original ratio, allowing for small rounding differences.

7. Why might the complement show “Not acute”?

A complement is usually discussed for acute angles. If the principal angle falls outside 0° to 90°, the calculator reports that an acute complement is not applicable.

8. What does the graph help me see?

The Plotly graph shows the corresponding forward trig function over its principal interval. Your solved point is highlighted, making inverse behavior easier to understand visually.