Inverse Trigonometric Functions
The inverse trigonometric functions arcsin, arccos and arctan give the angle that has a given sine, cosine or tangent, and each is defined on a restricted domain so that it returns a single value. In brief: arcsin(1/2) = π/6 = 30°, arccos(0) = π/2 = 90° and arctan(1) = π/4 = 45°. The ranges are [−π/2, π/2] for arcsin, [0, π] for arccos and (−π/2, π/2) for arctan.
Sine, cosine and tangent are periodic, so many angles share the same value, and the functions are not one-to-one on their full domains. Restricting the domain to an interval where the function is monotonic gives an inverse. This guide explains the restrictions, the graphs, the main identities, how to compute compositions such as sin(arccos x) and how to use the inverse functions to find angles, with a table of values.
Why do trigonometric functions need restricted domains?
A function has an inverse only if it is one-to-one, that is, every output comes from exactly one input. But sin x = 1/2 for x = π/6, 5π/6, 13π/6 and infinitely many others. To define an inverse, choose an interval on which the function covers all its values exactly once, and use that part of the graph.
What are arcsin, arccos and arctan?
- y = arcsin x (also written sin−1x) means sin y = x with y in [−π/2, π/2]. Domain: [−1, 1].
- y = arccos x (cos−1x) means cos y = x with y in [0, π]. Domain: [−1, 1].
- y = arctan x (tan−1x) means tan y = x with y in (−π/2, π/2). Domain: all real numbers, with horizontal asymptotes y = ±π/2.
The graph of each inverse function is the reflection of the restricted original in the line y = x. For instance, arcsin is the reflection of the part of sin x between −π/2 and π/2. Properties: arcsin and arctan are odd and increasing; arccos is decreasing and satisfies arccos(−x) = π − arccos x.
The notation sin−1x does not mean 1/sin x. The reciprocal is csc x. The functions arccot, arcsec and arccsc exist, but are less used.
How do you evaluate compositions?
The inverse relations hold only within the restricted ranges:
- sin(arcsin x) = x for x in [−1, 1], but arcsin(sin x) = x only for x in [−π/2, π/2].
- cos(arccos x) = x for x in [−1, 1], but arccos(cos x) = x only for x in [0, π].
- tan(arctan x) = x for all x, but arctan(tan x) = x only for x in (−π/2, π/2).
Example 1
arcsin(sin(5π/6)): sin(5π/6) = 1/2, and arcsin(1/2) = π/6. The answer is π/6, not 5π/6, since 5π/6 is outside the range of arcsin.
Example 2
cos(arcsin(3/5)): let θ = arcsin(3/5), so sin θ = 3/5 with θ in [−π/2, π/2], and cos θ ≥ 0 there. In a right triangle with opposite 3 and hypotenuse 5, the adjacent side is 4, so cos θ = 4/5. Algebraically: cos θ = √(1 − 9/25) = 4/5.
Example 3
tan(arccos(−1/2)): arccos(−1/2) = 2π/3, and tan(2π/3) = −√3.
Example 4
sin(2 arctan(3/4)): let θ = arctan(3/4), so the triangle has legs 3 and 4 and hypotenuse 5, and sin θ = 3/5, cos θ = 4/5. Then sin 2θ = 2 · (3/5)(4/5) = 24/25.
What identities do the inverse functions satisfy?
- arcsin x + arccos x = π/2 for x in [−1, 1].
- arcsin(−x) = −arcsin x, arctan(−x) = −arctan x.
- arccos(−x) = π − arccos x.
- arctan x + arctan(1/x) = π/2 for x > 0.
- sin(arccos x) = √(1 − x2) and cos(arcsin x) = √(1 − x2).
- tan(arcsin x) = x/√(1 − x2).
Check: arcsin(1/2) + arccos(1/2) = π/6 + π/3 = π/2 ✓. And arctan 2 + arctan(1/2) ≈ 1.1071 + 0.4636 = 1.5708 ✓.
Table of values
| x | arcsin x | arccos x | arctan x |
|---|---|---|---|
| −1 | −π/2 | π | −π/4 |
| −√3/2 | −π/3 | 5π/6 | – |
| −1/2 | −π/6 | 2π/3 | – |
| 0 | 0 | π/2 | 0 |
| 1/2 | π/6 | π/3 | – |
| √2/2 | π/4 | π/4 | – |
| √3/2 | π/3 | π/6 | – |
| 1 | π/2 | 0 | π/4 |
| √3 | not defined | not defined | π/3 |
A dash means that the value is not a simple multiple of π. For example, arctan(1/2) ≈ 0.4636 rad ≈ 26.57°.
How do you use them to find angles?
Example 5: a ramp
A ramp 5 m long reaches a height of 1 m. The angle is arcsin(1/5) = arcsin(0.2) ≈ 11.54°.
Example 6: the angle of a vector
The vector (−3, 3) makes the angle arctan(3/(−3)) = arctan(−1) = −45° with the positive x-axis according to the calculator. But the vector lies in quadrant II, so the true angle is 180° − 45° = 135°. A calculator’s arctan returns only values between −90° and 90°, so for points with a negative x-coordinate, add 180°. Programming languages provide atan2(y, x) to handle the quadrant automatically.
Example 7: the other solution in an equation
Solve sin x = 0.8 in [0°, 360°): arcsin 0.8 ≈ 53.13°, so x ≈ 53.13° and x ≈ 180° − 53.13° = 126.87°. The inverse function gives only the first; the second comes from the symmetry sin(180° − x) = sin x.
Where does the notation come from?
The names “arcsin” and similar are from the Latin arcus, because the value is the length of the arc on the unit circle with a given sine. The notation sin−1 was introduced by John Herschel in 1813, as part of a general notation for inverse functions, and it remains in use despite the possible confusion with the reciprocal. Inverse trigonometric functions were used earlier, for instance in the series expansions by Gregory and Leibniz for arctan, which led to the formula π/4 = 1 − 1/3 + 1/5 − 1/7 + …
Where are they used today?
Angles are computed from side lengths whenever a measured ratio is known: the slope angle of a roof, the elevation of a satellite dish, the angle of a robot’s joint. In calculus, arcsin and arctan appear as antiderivatives: the integral of 1/(1 + x2) is arctan x. The angle between two vectors is arccos of the normalised dot product, and computer graphics uses atan2 to turn coordinates into directions.
Which mistakes are common?
Reading sin−1 as a reciprocal
sin−1x = arcsin x, not 1/sin x.
Assuming arcsin(sin x) = x for every x
It holds only if x is in [−π/2, π/2]. Otherwise find the equivalent angle in the range.
Forgetting the range when giving the answer
arccos(−1/2) = 2π/3 only. The angle 4π/3 also has cosine −1/2, but it is outside [0, π].
Using the calculator’s arctan for quadrant II and III
Adjust the result by 180° if the x-coordinate is negative, or use atan2.