QuickCalculators evaluates arcsin, arccos, arctan, arccot, arcsec, and arccsc for a real input in range, returning the principal value in degrees or radians. Enter the ratio, choose the function and unit, and read the angle with its domain and principal range stated beside the result.
Find an angle from a trigonometric ratio
An inverse trigonometric function recovers an angle from a sine, cosine, tangent, or reciprocal ratio. Inverse Trigonometric Functions Calculator applies the principal branch of each inverse so every valid input maps to one standard angle. Forward trig of that principal value returns the original ratio within rounding.
Example: arcsin(0.5) = 30° or π/6 radians because sin(30°) = 0.5. The same ratio also equals sin(150°) and infinitely many coterminal and supplementary patterns, but the principal arcsin value is 30°. The page states that distinction in the misconception section below.
Read the domain and range of each function
Each inverse has a domain of allowed inputs and a principal range of returned angles. Arcsin and arccos require arguments in [-1, 1]; arctan accepts any real; arcsec and arccsc require |x| ≥ 1. QuickCalculators rejects out-of-domain values with a field error instead of inventing an angle.
Principal ranges in brief: arcsin returns [-90°, 90°], arccos returns [0°, 180°], and arctan returns (-90°, 90°) in degree mode. Matching radian intervals replace those degree endpoints. Knowing the range explains why two calculators should agree on the principal answer even when many angles share the same trig value.
Avoid this common mistake
Arcsin(0.5) = 30° is one of infinitely many correct angles whose sine is 0.5. The calculator returns the principal value, not the only value. Solving equations such as sin θ = 0.5 on a full circle still needs the general solution after the principal value is known.
General form in degrees for sine: θ = 30° + 360°k or θ = 150° + 360°k for integer k. Principal-value tools stop at 30° for arcsin(0.5). Homework that asks for all solutions must continue past that single reading.
Convert between degrees and radians
Degree and radian modes express the same principal angle in different units. Conversion uses π radians = 180°. Inverse Trigonometric Functions Calculator switches the displayed unit without changing which principal branch was chosen. So 30° equals π/6 rad, and 90° equals π/2 rad.
Entering the same ratio in either mode should produce matching measures after conversion. Mixing mode labels without converting is a common compare-the-answer error.
Frequently asked questions
What is an inverse trigonometric function?
An inverse trigonometric function returns an angle whose sine, cosine, tangent, or reciprocal equals a given real number, using a defined principal range. Examples include arcsin, arccos, and arctan. The calculator evaluates those principal values for valid inputs.
What is arcsin of 0.5?
Arcsin of 0.5 is 30 degrees, or π/6 radians, as the principal value. Many other angles also have sine 0.5, but arcsin itself returns that principal reading. Enter 0.5 in arcsin mode to confirm.
What is the range of arccos?
The range of arccos in degree mode is 0° to 180° inclusive. That principal interval picks one angle for each cosine in [-1, 1]. Radian mode uses 0 to π for the same branch.
Why does arcsin only accept values between -1 and 1?
Arcsin only accepts values between -1 and 1 because those are the only real outputs of the sine function. Inputs outside that interval have no real arcsin. The tool errors on such entries rather than returning a complex value.
What is the principal value?
The principal value is the single angle chosen from the inverse function's agreed range. It is the standard calculator output for that inverse. Other angles with the same trig ratio exist but are outside that principal selection.
How do you find all the angles, not just one?
To find all the angles, start from the principal value and apply the general solution for that trig equation, adding full rotations and any co-terminal or supplementary patterns the identity requires. The calculator supplies the principal seed for that longer list.
Summary
The calculator returns principal arcsin, arccos, arctan, arccot, arcsec, and arccsc values in degrees or radians. Arcsin(0.5) is 30° on the principal branch even though infinitely many angles share that sine. Domains such as [-1, 1] for arcsin are enforced. Degree and radian modes express the same principal angle after unit conversion.