This inverse trigonometric functions calculator will help you solve all calculations involving the inverse trigonometric functions. Pick a function (arcsin, arccos, arctan, arccot, arcsec, or arccsc), type in a value, and instantly get the resulting angle in degrees, radians, radians as a multiple of π, and gradians.
In this article, we'll help you understand what the inverse trigonometric functions are, what the inverse trigonometric functions ranges are, and how to calculate the inverse trigonometric functions. We will also demonstrate some practical examples to help you understand these concepts.
What are the trigonometric functions and their inverses?
Trigonometric functions are mathematical functions that help to describe the relationship between the sides of a right triangle. The most common trigonometric functions are the sine, cosine and tangent, and more rarely you may come across the cotangent, secant and cosecant.
The inverses of these functions are arcsine, arccosine, arctangent, arccotangent, arcsecant and arccosecant. The inverse trigonometric functions ranges and other useful information are summarized in the table below.
| Function | Domain | Range |
|---|---|---|
| arcsin(x) | −1 ≤ x ≤ 1 | −π/2 ≤ θ ≤ π/2 |
| arccos(x) | −1 ≤ x ≤ 1 | 0 ≤ θ ≤ π |
| arctan(x) | All real numbers | −π/2 < θ < π/2 |
| arccot(x) | All real numbers | 0 < θ < π |
| arcsec(x) | x ≤ −1 or x ≥ 1 | 0 ≤ θ < π/2 or π/2 < θ ≤ π |
| arccsc(x) | x ≤ −1 or x ≥ 1 | −π/2 ≤ θ < 0 or 0 < θ ≤ π/2 |
When applied to an angle, trigonometric functions return the ratio of the sides of a right triangle. So, in contrast, inverse trigonometric functions return the angle between two sides of a right triangle when they are applied to the ratio of these sides.
For instance, arcsin(x) returns the angle when applied to the ratio of the opposite side of the triangle to the hypotenuse, arccos(x) returns the respective angle when applied to the ratio of the adjacent side of the triangle to the hypotenuse, and arctan(x) returns the angle when applied to the ratio of the opposite side of the triangle to the adjacent side.
Now, let's look at how to calculate the inverse trigonometric functions.
How to calculate inverse trigonometric functions
Inverse trigonometric functions are used to calculate the angles in a right-angled triangle when the ratio of the sides adjacent to that angle is known. To understand both the concept and the calculation, let's look at how to calculate the arcsine for a right triangle.
- The inverse trigonometric function is arcsin (also denoted as sin⁻¹).
- The domain of arcsin (the range of valid inputs) is −1 ≤ x ≤ 1.
- The range of arcsin (the range of possible outputs) is −π/2 ≤ θ ≤ π/2.
- Suppose the opposite side's length is 2 cm for our example.
- Suppose the hypotenuse's length is 4 cm.
For this calculation, we would like to determine the angle (θ) with arcsin:
θ = arcsin(opposite / hypotenuse) = arcsin(2 / 4) = arcsin(1 / 2)
To calculate the inverse of the trigonometric function, use the formula:
sin(θ) = opposite / hypotenuse ⟹ sin(θ) = 2 / 4 = 1 / 2
We happen to know that sin(30°) = 1/2. Hence, θ = 30°. You can also display the answer in radians, which will be 30 × (π / 180) = 0.5236 rad. To understand more on this topic, please use our right triangle trigonometry resources.
Formulas used by this calculator
Most programming languages and calculators only provide arcsin, arccos and arctan directly. The remaining three inverse functions are computed from them using the reciprocal identities:
- arccot(x) = π/2 − arctan(x) (principal value in (0, π))
- arcsec(x) = arccos(1 / x)
- arccsc(x) = arcsin(1 / x)
Each result is a principal value: because the trigonometric functions are periodic, infinitely many angles share the same ratio, so the inverse is restricted to a single interval (shown in the table above). The Verification line re-applies the forward function to confirm the answer.
Common inverse-function values
| x | arcsin(x) | arccos(x) | arctan(x) |
|---|---|---|---|
| −1 | −90° | 180° | −45° |
| −√3/2 ≈ −0.8660 | −60° | 150° | −40.89° |
| −1/2 = −0.5 | −30° | 120° | −26.57° |
| 0 | 0° | 90° | 0° |
| 1/2 = 0.5 | 30° | 60° | 26.57° |
| √2/2 ≈ 0.7071 | 45° | 45° | 35.26° |
| √3/2 ≈ 0.8660 | 60° | 30° | 40.89° |
| 1 | 90° | 0° | 45° |
Example: How to use the inverse trigonometric functions calculator
- Select the inverse function you need — for example, arcsin.
- Enter the value in the input field. For example, type 0.5.
- Click Calculate.
- The calculator instantly shows the angle in degrees, radians, radians as a multiple of π, and gradians.
- For arcsin(0.5): the result is 30° = π/6 rad ≈ 0.5236 rad = 33.333 grad.
- The Verification line confirms that sin(30°) = 0.5 ✓
FAQs
What is the arcsin of 0.5?
The arcsin of 0.5 is 0.5236 radians or 30 degrees. The arcsin is the inverse of sin. Hence, if x = sin(θ), then θ = arcsin(x).
Why do arcsec and arccsc require |x| ≥ 1?
The secant and cosecant of a real angle always have an absolute value of at least 1 (since sec = 1/cos and csc = 1/sin, and cosine and sine never exceed 1 in magnitude). So there is no real angle whose secant is, say, 0.5 — arcsec(0.5) is undefined in the real numbers. Enter a value with |x| ≥ 1 for these two functions.
What is the difference between arccot from arctan?
This calculator uses the convention arccot(x) = π/2 − arctan(x), which returns a principal value in (0, π) — a continuous decreasing curve. Some textbooks instead define arccot on (−π/2, 0) ∪ (0, π/2]; be sure to check which convention your source uses.
How are degrees, radians, and gradians related?
360° = 2π rad = 400 grad. Degrees split a full circle into 360 parts; radians measure arc length on a unit circle; gradians divide a right angle into 100 equal parts (used in surveying). This calculator shows all three for easy conversion.
Is cos⁻¹(x) the same as 1/cos(x)?
No. The notation cos⁻¹(x) (and likewise sin⁻¹, tan⁻¹) means the inverse function — the angle whose cosine is x — not the reciprocal 1/cos(x). To avoid the confusion, the "arc" names (arccos, arcsin, arctan) are often preferred.