Graphs of inverse trigonometric functions
The graph of an inverse function is the mirror image of the principal branch in the line y = x: swap the x and y coordinates of every point.
Swap x and y
If a function takes a to b, its inverse takes b back to a. So every point (a, b) on the graph of the original function becomes the point (b, a) on the graph of the inverse. Swapping the coordinates is the same as reflecting in the line y = x. Only the principal branch is reflected: reflecting the whole sine wave would give a curve that fails the vertical line test, so it would not be a function.
Shapes of all six graphs
| Graph | Domain | Range | Passes through | Shape |
|---|---|---|---|---|
| y = sin⁻¹ x | [−1, 1] | [−π/2, π/2] | (−1, −π/2), (0, 0), (1, π/2) | Increasing, steep at the ends |
| y = cos⁻¹ x | [−1, 1] | [0, π] | (−1, π), (0, π/2), (1, 0) | Decreasing |
| y = tan⁻¹ x | R | (−π/2, π/2) | (−1, −π/4), (0, 0), (1, π/4) | Increasing, levels off near y = ±π/2 |
| y = cot⁻¹ x | R | (0, π) | (−1, 3π/4), (0, π/2), (1, π/4) | Decreasing, levels off near y = π and y = 0 |
| y = sec⁻¹ x | R − (−1, 1) | [0, π] − {π/2} | (−1, π), (1, 0) | Two pieces, each increasing, both approaching y = π/2 |
| y = cosec⁻¹ x | R − (−1, 1) | [−π/2, π/2] − {0} | (−1, −π/2), (1, π/2) | Two pieces, each decreasing, both approaching y = 0 |
How to draw y = sin x and y = sin⁻¹ x on the same axes
- 1Draw the x-axis and the y-axis. Mark −1 and 1 on both axes, and mark −π/2 and π/2 (about ±1.57) on both axes.
- 2Draw the line y = x as a dashed line and label it.
- 3Plot (−π/2, −1), (−π/6, −1/2), (0, 0), (π/6, 1/2) and (π/2, 1). Join them with a smooth curve and label it y = sin x.
- 4★Swap each point: (−1, −π/2), (−1/2, −π/6), (0, 0), (1/2, π/6), (1, π/2). Join them with a smooth curve that is steep near the ends, and label it y = sin⁻¹ x.
- 5Put solid dots at the end points. Write 'domain [−1, 1], range [−π/2, π/2]' beside the inverse curve.
★ key step
How to draw y = cos⁻¹ x and y = tan⁻¹ x
- cos⁻¹ x: mark π/2 and π on the y-axis. Plot (−1, π), (−1/2, 2π/3), (0, π/2), (1/2, π/3) and (1, 0). Join them with a smooth falling curve, with solid dots at both ends.
- tan⁻¹ x: draw dashed horizontal lines y = π/2 and y = −π/2 and label them. Plot (−√3, −π/3), (−1, −π/4), (0, 0), (1, π/4) and (√3, π/3).
- Join the tan⁻¹ points with a smooth rising S-shaped curve that gets closer to the dashed lines but never touches them.
- Label every curve and both axes. Examiners look for the key points and the labels.
Example 1: reading a graph
A curve passes through (−1, −π/4), (0, 0) and (1, π/4). It exists for every real x and levels off near y = π/2 and y = −π/2 without reaching them. Name the function and state its domain and range.
- Rule: compare the key points with the table of shapes.
- tan(π/4) = 1, so (1, π/4) fits y = tan⁻¹ x. (−1, −π/4) and (0, 0) fit as well.
- The curve exists for every real x, so the domain is R.
- It never reaches ±π/2, so the range is the open interval (−π/2, π/2).
It is y = tan⁻¹ x, with domain R and range (−π/2, π/2).
Example 2: using the mirror rule
The point (π/6, 1/2) lies on y = sin x. Find the matching point on y = sin⁻¹ x, and use it to write sin⁻¹(1/2).
- Rule: (a, b) on the principal branch of y = sin x gives (b, a) on y = sin⁻¹ x.
- π/6 lies in [−π/2, π/2], so (π/6, 1/2) is on the principal branch.
- Swap the coordinates: (1/2, π/6).
- So when x = 1/2, y = sin⁻¹(1/2) = π/6.
The point is (1/2, π/6), so sin⁻¹(1/2) = π/6.
The S-shaped tan⁻¹ curve turns up in robot arms and drones: their controllers use inverse tangent to turn sensor readings into a turning angle that always stays between −90° and 90°.
1 marks: “The graph of y = sin⁻¹ x is the mirror image of the principal branch of y = sin x in which line?” The line y = x.
2 marks: “Sketch the graph of y = cos⁻¹ x and write its domain and range.” Plot (−1, π), (0, π/2) and (1, 0), and join them with a smooth decreasing curve with solid dots at the ends. Domain [−1, 1], range [0, π].
Swap and reflect: (a, b) becomes (b, a), with y = x as the mirror. sin⁻¹ and tan⁻¹ rise; cos⁻¹ and cot⁻¹ fall.