Graphs and properties of inverse trig functions

Each inverse trigonometric function inherits properties from its parent. Monotonicity flips orientation only when the parent is decreasing, and the graph is always a reflection of the restricted parent across the line $y = x$. This subtopic catalogues the six graphs and the symmetries you should know on sight.

The six graphs

$y = \sin^{-1} x$. Domain $[-1, 1]$, range $[-\pi/2, \pi/2]$. Increasing. Passes through $(-1, -\pi/2), (0, 0), (1, \pi/2)$. Odd function: $\sin^{-1}(-x) = -\sin^{-1} x$.

$y = \cos^{-1} x$. Domain $[-1, 1]$, range $[0, \pi]$. Decreasing. Passes through $(-1, \pi), (0, \pi/2), (1, 0)$. Reflection of $\sin^{-1}$ about the line $y = \pi/4$: in fact $\cos^{-1} x = \pi/2 - \sin^{-1} x$.

$y = \tan^{-1} x$. Domain $\mathbb{R}$, range $(-\pi/2, \pi/2)$. Increasing. Horizontal asymptotes $y = \pm \pi/2$. Odd: $\tan^{-1}(-x) = -\tan^{-1} x$.

$y = \cot^{-1} x$. Domain $\mathbb{R}$, range $(0, \pi)$. Decreasing. Asymptotes $y = 0$ (as $x \to \infty$) and $y = \pi$ (as $x \to -\infty$). $\cot^{-1}(-x) = \pi - \cot^{-1} x$.

$y = \sec^{-1} x$. Domain $|x| \ge 1$, range $[0, \pi] \setminus \{\pi/2\}$. Two branches: one on $[1, \infty)$ increasing from $0$ to $\pi/2$, one on $(-\infty, -1]$ increasing from $\pi/2$ to $\pi$.

$y = \csc^{-1} x$. Domain $|x| \ge 1$, range $[-\pi/2, \pi/2] \setminus \{0\}$. Decreasing on each branch, symmetric to $\sec^{-1}$.

Symmetry identities

Monotonicity by inheritance

Since $\sin$ is increasing on $[-\pi/2, \pi/2]$, $\sin^{-1}$ is increasing on $[-1, 1]$. Since $\cos$ is decreasing on $[0, \pi]$, $\cos^{-1}$ is decreasing on $[-1, 1]$. The same logic applies to the other four.

Derivatives (preview)

The derivatives are computed in Chapter 5, but it is useful to know:

Reading the graph: an example

Suppose you are asked: for which $x$ is $\sin^{-1} x = \tan^{-1} x$? From the graphs, both start at $0$ for $x = 0$ and both are increasing. $\sin^{-1}$ rises faster (its derivative $1/\sqrt{1 - x^2}$ exceeds $1/(1 + x^2)$ for $x \in (0, 1)$). So the only intersection is at $x = 0$.

Worked examples

Example 1. Without computing, decide whether $\sin^{-1}(0.6) > \sin^{-1}(0.5)$.

Yes, because $\sin^{-1}$ is strictly increasing on $[-1, 1]$.

Example 2. Decide whether $\cos^{-1}(0.4) > \cos^{-1}(0.5)$.

Yes, because $\cos^{-1}$ is strictly decreasing.

Example 3. Sketch $y = \sin^{-1}(\sin x)$ for $x \in [-2\pi, 2\pi]$.

Inside $[-\pi/2, \pi/2]$, the function is just $x$. Outside, fold using $\sin(\pi - x) = \sin x$. The graph is a triangular wave with peaks at $\pm \pi/2$ and slope alternating between $+1$ and $-1$.

Example 4. Sketch $y = \cos^{-1}(\cos x)$ on $[-2\pi, 2\pi]$.

Inside $[0, \pi]$, it is $x$. Use $\cos(-x) = \cos x$ and periodicity to fold. The graph is a triangular wave with peaks at $\pi$ and troughs at $0$, $2\pi$.

Example 5. Show that $\sin^{-1} x + \cos^{-1} x = \pi/2$ for all $x \in [-1, 1]$.

Let $\theta = \sin^{-1} x$, so $x = \sin \theta$ and $\theta \in [-\pi/2, \pi/2]$. Then $\cos(\pi/2 - \theta) = \sin \theta = x$, and $\pi/2 - \theta \in [0, \pi]$. So $\cos^{-1} x = \pi/2 - \theta$, giving the identity.

Example 6. Plot $\tan^{-1} x$ and $\cot^{-1} x$ on the same axes. Confirm that $\tan^{-1} x + \cot^{-1} x = \pi/2$ for all $x$.

The two curves are reflections of each other across $y = \pi/4$, summing to a constant $\pi/2$ at every $x$.

Try it yourself

1. Sketch $y = \sin^{-1} x$ and mark its monotonicity.
2. Find $\cos^{-1}(\cos(2.5))$ given $\pi \approx 3.14$.
3. Show $\sin^{-1}(-x) = -\sin^{-1} x$ using the principal branch.
4. Why is the graph of $\cot^{-1} x$ continuous at $x = 0$ but not differentiable in elementary ways?
5. Sketch $y = \sin^{-1} x + \cos^{-1} x$ for $x \in [-1, 1]$ , note the constant value.
6. Compute $\sec^{-1}(\sqrt 2)$.
7. Compute $\sec^{-1}(-\sqrt 2)$.
8. Find the points where $y = \sin^{-1}(\sin x)$ has its peaks for $x \in [-3\pi, 3\pi]$.
9. Sketch $y = \tan^{-1}(\tan x)$ for $x \in [-\pi, \pi]$.
10. State the horizontal asymptotes of $\tan^{-1} x$.
11. Why does $\cos^{-1}$ have no points of inflection?
12. Verify the identity $\tan^{-1} x + \cot^{-1} x = \pi/2$ by differentiating both sides.
13. For which $x$ is $\sin^{-1} x = \cos^{-1} x$?
14. Show that $\sin^{-1} x + \sin^{-1}(-x) = 0$ for $x \in [-1, 1]$.

Pitfalls / Tricks

• $\cos^{-1}$ is not odd. $\cos^{-1}(-x) = \pi - \cos^{-1} x$, not $-\cos^{-1} x$.
• The graph of $\sin^{-1}(\sin x)$ is piecewise linear , a sawtooth, not a smooth sine.
• Conventions for $\sec^{-1}$ vary; in some books the range includes $[\pi/2, \pi)$ on the left branch instead of $(\pi/2, \pi]$.
• The derivative of $\sin^{-1} x$ blows up at $x = \pm 1$ , the graph has vertical tangent at the endpoints.

Next we collect the standard inverse trigonometric identities.