Inverse Functions: Arcsecant (Part 29)

We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of y = \sec x satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of y = \sec^{-1} x uses the interval [0,\pi] — or, more precisely, [0,\pi/2) \cup (\pi/2, \pi] to avoid the vertical asymptote at x = \pi/2 — in order to approximately match the range of \cos^{-1} x. However, when I was a student, I distinctly remember that my textbook chose [0,\pi/2) \cup [\pi,3\pi/2) as the range for \sec^{-1} x.

I believe that this definition has fallen out of favor today. However, for the purpose of today’s post, let’s just run with this definition and see what happens. This portion of the graph of y = \sec x is perhaps unorthodox, but it satisfies the horizontal line test so that the inverse function can be defined.

arcsec3

Let’s fast-forward a couple of semesters and use implicit differentiation (see also https://meangreenmath.com/2014/08/08/different-definitions-of-logarithm-part-8/ for how this same logic is used for other inverse functions) to find the derivative of y = \sec^{-1} x:

x = \sec y

\displaystyle \frac{d}{dx} (x) = \displaystyle \frac{d}{dx} (\sec y)

1 = \sec y \tan y \displaystyle \frac{dy}{dx}

\displaystyle \frac{1}{\sec y \tan y} = \displaystyle \frac{dy}{dx}

 At this point, the object is to convert the left-hand side to something involving only x. Clearly, we can replace \sec y with x. As it turns out, the replacement of \tan y is a lot simpler than we saw in yesterday’s post. Once again, we begin with one of the Pythagorean identities:

1 + \tan^2 y = \sec^2 y

\tan^2 y = \sec^2 y - 1

\tan^2 y = x^2 - 1

\tan y = \sqrt{x^2 - 1} \qquad \hbox{or} \tan y = -\sqrt{x^2 - 1}

So which is it, the positive answer or the negative answer? In yesterday’s post, the answer depended on whether x was positive or negative. However, with the current definition of \sec^{-1} x, we know for certain that the answer is the positive one! How can we be certain? The angle y must lie in either the interval [0,\pi/2) or else the interval [\pi,3\pi/2). In either interval, \tan y is positive. So, using this definition of \sec^{-1} x, we can simply say that

\displaystyle \frac{d}{dx} \sec^{-1} x = \displaystyle \frac{1}{x \sqrt{x^2-1}},

and we don’t have to worry about |x| that appeared in yesterday’s post.

green line

arcsec2Turning to integration, we now have the simple formula

\displaystyle \int \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} x + C

that works whether x is positive or negative. For example, the orange area can now be calculated correctly:

\displaystyle \int_{-2}^{-2\sqrt{3}/3} \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} \left( - \displaystyle \frac{2\sqrt{3}}{3} \right) - \sec^{-1} (-2)

= \displaystyle \frac{7\pi}{6} - \frac{4\pi}{3}

= \displaystyle -\frac{\pi}{6}

So, unlike yesterday’s post, this definition of \sec^{-1} x produces a simple integration formula.

green line

So why isn’t this the standard definition for \sec^{-1} x? I’m afraid the answer is simple: with this definition, the equation

\sec^{-1} x = \cos^{-1} \left( \displaystyle \frac{1}{x} \right)

is no longer correct if x < -1. Indeed, I distinctly remember thinking, back when I was a student taking trigonometry, that the definition of \sec^{-1} x seemed really odd, and it seemed to me that it would be better if it matched that of \cos^{-1} x. Of course, at that time in my mathematical development, it would have been almost hopeless to explain that the range [0,\pi/2) \cup [\pi,3\pi/2) had been chosen to simplify certain integrals from calculus.

So I suppose that The Powers That Be have decided that it’s more important for this identity to hold than to have a simple integration formula for \displaystyle \int \frac{dx}{x \sqrt{x^2 -1}}

Inverse Functions: Arcsecant (Part 28)

We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of y = \sec x satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of y = \sec^{-1} x uses the interval [0,\pi], so that

\sec^{-1} x = \cos^{-1} \left( \displaystyle \frac{1}{x} \right)

Why would this be controversial? Yesterday, we saw that \tan x is both positive and negative on the interval [0,\pi], and so great care has to be used to calculate the integral:

\displaystyle \int \frac{dx}{x \sqrt{x^2 -1}}

Here’s another example: let’s use trigonometric substitution to calculate

\displaystyle \int_{-6}^{-3} \frac{ \sqrt{x^2-9} }{x} dx

The standard trick is to use the substitution x = 3 \sec \theta. With this substitution:

  • x^2 - 9 = 9 \sec^2 \theta - 9 = 9 \tan^2 \theta, and
  • dx = 3 \sec \theta \tan \theta \, d\theta
  • x = -3 \quad \Longrightarrow \quad \sec \theta = -1 \quad \Longrightarrow \quad \theta = \sec^{-1} (-1) = \pi
  • x = -6 \quad \Longrightarrow \quad \sec \theta = -2 \quad \Longrightarrow \quad \theta = \sec^{-1} (-2) = \displaystyle \frac{2\pi}{3}

Therefore,

\displaystyle \int_{-6}^{-3} \frac{ \sqrt{x^2-9} }{x} dx = \displaystyle \int_{2\pi/3}^{\pi} \frac{ \sqrt{9 \tan^2 \theta}} { 3 \sec \theta} 3 \sec \theta \tan \theta \, d\theta

At this point, the common mistake would be to replace \sqrt{9 \tan^2 \theta} with 3 \tan \theta. This is a mistake because

\sqrt{9 \tan^2 \theta} = |3 \tan \theta|

Furthermore, for this particular problem, \tan \theta is negative on the interval [2\pi/3,\pi]. Therefore, for this problem, we should replace \sqrt{9 \tan^2 \theta} with -3 \tan \theta.

Continuing the calculation,

\displaystyle \int_{-6}^{-3} \frac{ \sqrt{x^2-9} }{x} dx = \displaystyle \int_{2\pi/3}^{\pi} \frac{ -3 \tan \theta} { 3 \sec \theta} 3 \sec \theta \tan \theta

= \displaystyle \int_{2\pi/3}^{\pi} -3\tan^2 \theta \, d\theta

= \displaystyle \int_{2\pi/3}^{\pi} -3(\sec^2 \theta-1) \, d\theta

= \displaystyle \int_{2\pi/3}^{\pi} (3-3 \sec^2 \theta) \, d\theta

= \displaystyle \bigg[ 3 \theta - 3 \tan \theta \bigg]_{2\pi/3}^{\pi}

= \displaystyle \left[ 3 \pi - 3 \tan \pi \right] - \left[ 3 \left( \frac{2\pi}{3} \right) - 3 \tan \left( \frac{2\pi}{3} \right) \right]

= \displaystyle 3\pi - 0 - 2\pi + 3(-\sqrt{3})

= \pi - 3\sqrt{3}

So if great care wasn’t used to correctly simplify \sqrt{9 \tan^2 \theta}, one would instead obtain the incorrect answer 3\sqrt{3} - \pi.

Inverse Functions: Arcsecant (Part 27)

We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of y = \sec x satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of y = \sec^{-1} x uses the interval [0,\pi], so that

\sec^{-1} x = \cos^{-1} \left( \displaystyle \frac{1}{x} \right)

Why would this be controversial? Let’s fast-forward a couple of semesters and use implicit differentiation (see also https://meangreenmath.com/2014/08/08/different-definitions-of-logarithm-part-8/ for how this same logic is used for other inverse functions) to find the derivative of y = \sec^{-1} x:

x = \sec y

\displaystyle \frac{d}{dx} (x) = \displaystyle \frac{d}{dx} (\sec y)

1 = \sec y \tan y \displaystyle \frac{dy}{dx}

\displaystyle \frac{1}{\sec y \tan y} = \displaystyle \frac{dy}{dx}

 At this point, the object is to convert the left-hand side to something involving only x. Clearly, we can replace \sec y with x. However, doing the same with \tan y is trickier. We begin with one of the Pythagorean identities:

1 + \tan^2 y = \sec^2 y

\tan^2 y = \sec^2 y - 1

\tan^2 y = x^2 - 1

\tan y = \sqrt{x^2 - 1} \qquad \hbox{or} \tan y = -\sqrt{x^2 - 1}

So which is it, the positive answer or the negative answer? The answer is, without additional information, we don’t know!

  • If 0 \le y < \pi/2 (so that x = \sec y \ge 1), then \tan y is positive, and so \tan y = \sqrt{\sec^2 y - 1}.
  • If \pi/2 < y \le \pi (so that x = \sec y \le 1), then \tan y is negative, and so \tan y = -\sqrt{\sec^2 y - 1}.

We therefore have two formulas for the derivative of y = \sec^{-1} x:

\displaystyle \frac{d}{dx} sec^{-1} x = \displaystyle \frac{1}{x \sqrt{x^2-1}} \qquad \hbox{if} \qquad x > 1

\displaystyle \frac{d}{dx} sec^{-1} x = \displaystyle -\frac{1}{x \sqrt{x^2-1}} \qquad \hbox{if} \qquad x < 1

These may then be combined into the single formula

\displaystyle \frac{d}{dx} sec^{-1} x = \displaystyle \frac{1}{|x| \sqrt{x^2-1}}

green lineIt gets better. Let’s now find the integral

\displaystyle \int \frac{dx}{x \sqrt{x^2 -1}}

arcsec2

Several calculus textbooks that I’ve seen will lazily give the answer

\displaystyle \int \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} x + C

This answer works as long as x > 1, so that |x| reduces to simply x. For example, the red signed area in the above picture on the interval [2\sqrt{3}/3,2] may be correctly computed as

\displaystyle \int_{2\sqrt{3}/3}^2 \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} 2 - \sec^{-1} \displaystyle \frac{2\sqrt{3}}{3}

= \cos^{-1} \left( \displaystyle \frac{1}{2} \right) - \cos^{-1} \left( \displaystyle \frac{\sqrt{3}}{2} \right)

= \displaystyle \frac{\pi}{3} - \frac{\pi}{6}

= \displaystyle \frac{\pi}{6}

However, the orange signed area on the interval [-2,-2\sqrt{3}/3]  is incorrectly computed using this formula!

\displaystyle \int_{-2}^{-2\sqrt{3}/3} \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} \left( - \displaystyle \frac{2\sqrt{3}}{3} \right) - \sec^{-1} (-2)

= \cos^{-1} \left( -\displaystyle \frac{\sqrt{3}}{2} \right) -\cos^{-1} \left( \displaystyle \frac{1}{2} \right)

= \displaystyle \frac{5\pi}{6} - \frac{2\pi}{3}

= \displaystyle \frac{\pi}{6}

This is patently false, as the picture clearly indicates that the above integral has to be negative. For this reason, careful calculus textbooks will often ask students to solve a problem like

\displaystyle \int \frac{dx}{x \sqrt{x^2 -1}}, \qquad x > 1

and the caveat x > 1 is needed to ensure that the correct antiderivative is used. Indeed, a calculus textbook that doesn’t include such caveats are worthy of any scorn that an instructor cares to heap upon it.

Inverse Functions: Arcsecant (Part 26)

We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of y = \sec x satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of y = \sec^{-1} x uses the interval [0,\pi] — or, more precisely, [0,\pi/2) \cup (\pi/2, \pi] to avoid the vertical asymptote at x = \pi/2. This portion of the graph of y = \sec x satisfies the horizontal line test and, conveniently, matches almost perfectly the domain of y = \cos^{-1} x. This is perhaps not surprising since, when both are defined, \cos x and \sec x are reciprocals.

arcsec1

 

Since this range of \sec^{-1} x matches that of \cos^{-1} x, we have the convenient identity

\sec^{-1} x = \cos^{-1} \left( \displaystyle \frac{1}{x} \right)

To see why this works, let’s examine the right triangle below. Notice that

\cos \theta = \displaystyle \frac{x}{1} \qquad \Longrightarrow \qquad \theta = \cos^{-1} x.

Also,

\sec\theta = \displaystyle \frac{1}{x} \qquad \Longrightarrow \qquad \theta = \cos^{-1} \left( \displaystyle \frac{1}{x} \right).

This argument provides the justification for 0 < \theta < \pi/2 — that is, for x > 1 — but it still works for x = 1 and x \le -1.

So this seems like the most natural definition in the world for \sec^{-1} x. Unfortunately, there are consequences for this choice in calculus, as we’ll see in tomorrow’s post.