Graphs of inverse trigonometric functions

arcsin, arccos and arctan undo sin, cos and tan on a restricted domain, so each graph is a reflection in y = x of one branch of the original. The ranges here are the principal values your calculator gives.

Every value in the key-features tables is checked by script against the Grapher's own calculations (roots, turning points, asymptotes, domain and range).

Graph of y = arcsin x

The inverse of sin x on −π/2 ≤ x ≤ π/2: the sine graph reflected in y = x, cut down so it is a function.

Open y = arcsin x in the Grapher

Key features

Domain	−1 ≤ x ≤ 1
Range	−π/2 ≤ y ≤ π/2
x-intercept	(0, 0)
y-intercept	(0, 0)
Turning points	none
Point of inflection	(0, 0)
Asymptotes	none
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercept (0, 0), the y-intercept (0, 0).
2. Only draw the curve where it is defined: −1 ≤ x ≤ 1.
3. Check the range: −π/2 ≤ y ≤ π/2. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

y = arcsin x is the reflection of y = sin x (for −π/2 ≤ x ≤ π/2) in the line y = x: see the graph of y = sin x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = arccos x

The inverse of cos x on 0 ≤ x ≤ π: it falls from (−1, π) to (1, 0).

Open y = arccos x in the Grapher

Key features

Domain	−1 ≤ x ≤ 1
Range	0 ≤ y ≤ π
x-intercept	(1, 0)
y-intercept	(0, π/2)
Turning points	none
Point of inflection	(0, π/2)
Asymptotes	none
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercept (1, 0), the y-intercept (0, π/2).
2. Only draw the curve where it is defined: −1 ≤ x ≤ 1.
3. Check the range: 0 ≤ y ≤ π. Your sketch should not go anywhere else.

y = arccos x is the reflection of y = cos x (for 0 ≤ x ≤ π) in the line y = x: see the graph of y = cos x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = arctan x

The inverse of tan x: defined for every x, with horizontal asymptotes y = ±π/2.

Open y = arctan x in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	−π/2 < y < π/2
x-intercept	(0, 0)
y-intercept	(0, 0)
Turning points	none
Point of inflection	(0, 0)
Asymptotes	horizontal: y = −π/2, as x → −∞; horizontal: y = π/2, as x → ∞
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Draw the asymptotes first as dashed lines: y = −π/2, as x → −∞; y = π/2, as x → ∞.
2. Mark the x-intercept (0, 0), the y-intercept (0, 0).
3. The domain is every real number, so the curve continues off both sides of your axes.
4. Check the range: −π/2 < y < π/2. Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

y = arctan x is the reflection of y = tan x (for −π/2 < x < π/2) in the line y = x: see the graph of y = tan x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

The Grapher is our own free tool (graph.mathrevision.co.uk), built for IB, A Level and IGCSE students. It is not an exam calculator: practise on your own GDC for calculator papers.