What is arcsin, arccos, and arctan?

Arcsin, arccos, and arctan (also written sin-1, cos-1, tan-1) are inverse trigonometric functions. They answer the question: which angle produces this trig ratio? - arcsin(x): finds the angle whose sine is x - arccos(x): finds the angle whose cosine is x - arctan(x): finds the angle whose tangent is x Examples: - arcsin(0.5) = 30 degrees, because sin(30) = 0.5 - arccos(0.5) = 60 degrees, because cos(60) = 0.5 - arctan(1) = 45 degrees, because tan(45) = 1 - arcsin(1) = 90 degrees - arctan(sqrt(3)) = 60 degrees Applications: finding angles in triangles from side lengths, solving trig equations, navigation and physics problems involving angles.

Why does arcsin only return values from -90 to 90 degrees?

Sine is not a one-to-one function — many different angles have the same sine value. For example, sin(30) = sin(150) = 0.5. For arcsin to be a proper function (each input gives exactly one output), we must restrict the output to a specific range. The principal value range for arcsin is -90 to 90 degrees (-pi/2 to pi/2). This range was chosen because: - It covers all possible sine values (-1 to 1) exactly once - It includes the most natural angles (near 0) - It keeps the function continuous Similarly: - arccos range: 0 to 180 degrees (0 to pi) - arctan range: -90 to 90 degrees (-pi/2 to pi/2) When solving trig equations, remember there may be additional solutions. If arcsin(x) = 30, the other angle with the same sine is 180 - 30 = 150 degrees. This calculator shows the alternative angle below the main result.

What is the identity arcsin(x) + arccos(x) = pi/2?

For any x between -1 and 1, arcsin(x) + arccos(x) = 90 degrees (pi/2 radians). This is because sine and cosine are complementary functions. Proof: - If arcsin(x) = theta, then sin(theta) = x - cos(90 - theta) = sin(theta) = x (co-function identity) - So arccos(x) = 90 - theta - Therefore arcsin(x) + arccos(x) = theta + (90 - theta) = 90 Verification: - arcsin(0.5) = 30, arccos(0.5) = 60. Sum = 90 degrees. - arcsin(sqrt(3)/2) = 60, arccos(sqrt(3)/2) = 30. Sum = 90 degrees. - arcsin(1) = 90, arccos(1) = 0. Sum = 90 degrees. This identity lets you find arccos from arcsin and vice versa: arccos(x) = 90 - arcsin(x).

How do I use arctan to find an angle in a triangle?

In a right triangle, if you know the opposite and adjacent sides, the angle is arctan(opposite/adjacent). Example: a right triangle with opposite side 3 and adjacent side 4: - tan(theta) = opposite/adjacent = 3/4 = 0.75 - theta = arctan(0.75) = 36.87 degrees If you know opposite and hypotenuse (sine): theta = arcsin(opposite/hypotenuse) If you know adjacent and hypotenuse (cosine): theta = arccos(adjacent/hypotenuse) arctan2 (atan2): programming languages often provide atan2(y, x) which gives the angle for a point (x, y) in all four quadrants, returning values from -180 to 180 degrees. This avoids the ambiguity of arctan which only gives -90 to 90. Use atan2 when you need the full-circle angle. For a point (x, y): angle = atan2(y, x) in most languages.

What is the difference between arctan and atan2?

arctan(x) computes the angle whose tangent is x, returning a value from -90 to 90 degrees. It takes a single number (the ratio of y to x). atan2(y, x) computes the angle for a point (x, y) in the full coordinate plane, returning -180 to 180 degrees. It takes two separate arguments and handles all four quadrants correctly. Why atan2 is needed: - arctan(1) = 45 degrees, which corresponds to both (1,1) in Quadrant I and (-1,-1) in Quadrant III - atan2(1, 1) = 45 degrees (Quadrant I) - atan2(-1, -1) = -135 degrees (Quadrant III) - arctan cannot distinguish these In programming: Python math.atan2(y, x), JavaScript Math.atan2(y, x), C atan2(y, x) In maths: atan2 is not used in this calculator since it requires two inputs. Use the Sin Cos Tan Calculator if you need to compute trig from coordinates.

Arcsin Arccos Arctan Calculator — Inverse Trig Functions

The Inverse Trig Calculator computes arcsin, arccos, and arctan for any valid input. Select your function, enter a value, and get the angle in degrees, radians, and gradians instantly. Exact radian expressions (pi/6, pi/4, pi/3) are shown for common inputs. The alternative angle ...

INVERSE TRIG CALCULATOR

Domain: −1 ≤ x ≤ 1Range: −90° to 90° (−π/2 to π/2)

Value for arcsin(x)

COMMON VALUES

arcsin(0.5) =

30°

DEGREES

30°

RADIANS

0.52359878

GRADIANS

33.333333

Alternative angle with the same sine: 150° (principal value is 30°)

Input: 0.5

arcsin(0.5) = 30°

= 0.5235987756 radians (π/6)

= 33.333333 gradians

Alternative angle with same sine: 150°

FUNCTION GRAPH

DOMAIN & RANGE REFERENCE

Function	Domain	Range
arcsin	−1 to 1	−90° to 90°
arccos	−1 to 1	0° to 180°
arctan	all reals	−90° to 90°

EXACT VALUES

x	arcsin(x)	arccos(x)	arctan(x)
0	0°	90°	0°
1/2	30°	60°	—
√2/2	45°	45°	—
√3/2	60°	30°	—
1	90°	0°	45°
−1/2	−30°	120°	—
−1	−90°	180°	—
√3	—	—	60°

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FORMULAS & IDENTITIES

INVERSE TRIG REFERENCE

NAME	FORMULA	DESCRIPTION
arcsin definition	arcsin(x) = theta where sin(theta) = x	Principal value in -pi/2 to pi/2
arccos definition	arccos(x) = theta where cos(theta) = x	Principal value in 0 to pi
arctan definition	arctan(x) = theta where tan(theta) = x	Principal value in -pi/2 to pi/2
Complementary	arcsin(x) + arccos(x) = pi/2	For any x in -1 to 1
Negative input	arcsin(-x) = -arcsin(x)	arcsin is an odd function
Negative input cos	arccos(-x) = pi - arccos(x)	arccos reflected about pi/2
arctan addition	arctan(x) + arctan(1/x) = pi/2 (x > 0)	Complementary arctan identity
Degrees conversion	degrees = radians x 180 / pi	Convert radian result to degrees
Sine of arcsin	sin(arcsin(x)) = x for x in -1 to 1	Composition identity
Arcsin of sine	arcsin(sin(x)) = x only for x in -pi/2 to pi/2	Not true for all x due to periodicity

HOW TO USE

1
Select the inverse function: arcsin (sin inverse), arccos (cos inverse), or arctan (tan inverse). The domain and range for each function are shown below the selector.
2
Enter a value. For arcsin and arccos, the input must be between -1 and 1. For arctan, any real number is valid. Use the common value buttons (0, 1/2, sqrt(2)/2, sqrt(3)/2, 1) for the standard inputs.
3
The angle appears instantly in the selected unit (degrees, radians, or gradians). Switch units with the buttons below the input — the result updates immediately without re-entering.
4
The alternative angle panel shows the other solution in the same period. For example, both 30 and 150 have the same sine (0.5), but arcsin returns only 30 (the principal value).
5
Click Show all three results to see arcsin, arccos, and arctan of the same input simultaneously in one table — useful when solving trig equations where you need to check all three functions.

WORKED EXAMPLE

arcsin(0.5)=30 degrees=pi/6 rad. arccos(0.5)=60 degrees=pi/3 rad. arcsin(0.5)+arccos(0.5)=90 degrees (identity). arctan(1)=45 degrees. arcsin(sqrt(3)/2)=60 degrees. arccos(-1)=180 degrees. arcsin(-1)=-90 degrees. arctan(sqrt(3))=60 degrees.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.