Inverse Trig Functions Calculator

Calculator Input

Inverse function

Input value x

Preferred output

Decimal precision

Integer cycle k

Step display

Show calculation steps

Formula Used

The calculator finds the principal inverse trigonometric value. It also shows repeating solution forms for full trigonometric equations.

Function	Formula	Domain	Principal range
sin⁻¹(x) / arcsin(x)	y = sin⁻¹(x), so sin(y) = x	-1 ≤ x ≤ 1	[-π/2, π/2]
cos⁻¹(x) / arccos(x)	y = cos⁻¹(x), so cos(y) = x	-1 ≤ x ≤ 1	[0, π]
tan⁻¹(x) / arctan(x)	y = tan⁻¹(x), so tan(y) = x	All real numbers	(-π/2, π/2)
cot⁻¹(x) / arccot(x)	y = cot⁻¹(x), so cot(y) = x	All real numbers	(0, π)
sec⁻¹(x) / arcsec(x)	y = sec⁻¹(x) = cos⁻¹(1/x)	x ≤ -1 or x ≥ 1	[0, π], y ≠ π/2
csc⁻¹(x) / arccsc(x)	y = csc⁻¹(x) = sin⁻¹(1/x)	x ≤ -1 or x ≥ 1	[-π/2, π/2], y ≠ 0

For secant and cosecant, the calculator first uses the reciprocal. Then it applies arccosine or arcsine.

How to Use This Calculator

Select the inverse trigonometric function.
Enter the input value x.
Choose the preferred output format.
Set decimal precision from 0 to 12.
Enter an integer k to inspect repeating solutions.
Press calculate to view the result above the form.
Use the CSV or PDF buttons to save the result.

Example Data Table

Input	Inverse function	Principal radians	Principal degrees	Check
0.5	sin⁻¹(x)	π/6	30°	sin(π/6) = 0.5
0.5	cos⁻¹(x)	π/3	60°	cos(π/3) = 0.5
1	tan⁻¹(x)	π/4	45°	tan(π/4) = 1
1	cot⁻¹(x)	π/4	45°	cot(π/4) = 1
2	sec⁻¹(x)	π/3	60°	sec(π/3) = 2
2	csc⁻¹(x)	π/6	30°	csc(π/6) = 2

Inverse Trig Functions Guide

Why inverse trig matters

Inverse trigonometric functions reverse normal trigonometric ratios. They answer a simple question. Which angle gives this sine, cosine, tangent, cotangent, secant, or cosecant value? This matters in geometry, vectors, surveying, navigation, electronics, waves, and analytic modeling. A calculator helps because inverse ratios have strict domains. Some values are valid for one function, yet invalid for another. The result also depends on the chosen principal range.

Principal angle meaning

Each inverse function returns one main angle. That angle is called the principal value. For arcsine, the answer stays between negative one half pi and positive one half pi. For arccosine, it stays from zero to pi. Arctangent and arccotangent use ranges that avoid repeated answers. Arcsecant and arccosecant use reciprocal rules, then follow cosine or sine ranges. This single value makes the function predictable.

Domain checks

Domain rules protect the calculation. Arcsine and arccosine accept only numbers from negative one to positive one. Arcsecant and arccosecant accept numbers less than or equal to negative one, or greater than or equal to positive one. Arctangent and arccotangent accept every real number. When the input is outside the valid interval, there is no real principal angle.

Using radians and degrees

Radians are standard in advanced mathematics. Degrees are often easier for practical work. This calculator shows both. It also gives a multiple of pi when a simple fraction is detected. That helps with exact angles, such as thirty degrees, forty five degrees, and sixty degrees. Decimal rounding can be adjusted for homework or reporting.

Checking the answer

A good inverse trigonometric answer should be verified. The check applies the original trig function to the computed angle. If the output returns the entered value, the calculation is consistent. For secant and cosecant, the calculator checks reciprocal cosine or sine. General solution forms are also shown for full equations. They include integer cycles, because trig functions repeat forever.

Common mistakes

Do not confuse inverse sine with reciprocal sine. The notation can look similar, but the meanings are different. Also check calculator mode before copying results. A radian answer placed into a degree formula can cause large errors. Keep the selected principal range in mind during final review.

FAQs

What does an inverse trig function calculate?

It calculates the angle that produces a given trigonometric ratio. For example, arcsine finds the angle whose sine equals the entered value.

Why do some inputs show a domain error?

Some inverse functions accept only certain values. Arcsine and arccosine need inputs from -1 to 1. Arcsecant and arccosecant need values outside that interval.

Does this calculator show radians and degrees?

Yes. It can show radians, degrees, or both. The result table also includes a pi-based form when a simple match is found.

Is inverse sine the same as cosecant?

No. Inverse sine returns an angle. Cosecant is the reciprocal of sine. Their notation can look similar, so the calculator labels each option clearly.

What is a principal value?

The principal value is the standard single angle returned by an inverse function. It stays inside the accepted range for that function.

Why are general solutions included?

Trigonometric functions repeat. A full equation can have many angles. The general solution shows how repeated answers appear through integer cycles.

Can I use decimal values?

Yes. Decimal and scientific notation values are accepted. The calculator validates the domain before calculating the angle.

What does the check value mean?

The check value applies the original trigonometric function to the computed angle. It should match the entered input after rounding.