Which of the following statements about the curve y = cosec^-1 x are true? I. It passes through the point (-1, -π/2). II. It passes through the point (2, π/3). III. It falls as x increases on [1, ∞). IV. It falls as x increases on (-∞, -1].

Question 77

MediumOriginal question

Which of the following statements about the curve $y = \operatorname{cosec}^{-1} x$ are true?
I. It passes through the point $\left(-1, -\frac{\pi}{2}\right)$ .
II. It passes through the point $\left(2, \frac{\pi}{3}\right)$ .
III. It falls as $x$ increases on $[1, \infty)$ .
IV. It falls as $x$ increases on $(-\infty, -1]$ .

All four: I, II, III and IVCorrect answerYour answerI and III onlyCorrect answerYour answerIII and IV onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\operatorname{cosec}\left(-\frac{\pi}{2}\right) = -1$ , so I is true; $\operatorname{cosec} \frac{\pi}{3} = \frac{2}{\sqrt{3}}$ , not 2, so II is false.
$\operatorname{cosec}^{-1} x = \sin^{-1} \frac{1}{x}$ : on each piece $\frac{1}{x}$ falls as $x$ increases and $\sin^{-1}$ keeps that order, so the curve falls, from $\frac{\pi}{2}$ towards 0 and from near 0 down to $-\frac{\pi}{2}$ . III and IV are true.

Worth noting

II mixes up $\operatorname{cosec}^{-1}$ with $\sec^{-1}$ : $\sec^{-1} 2 = \frac{\pi}{3}$ , but $\operatorname{cosec}^{-1} 2 = \frac{\pi}{6}$ .

Concept: Rising, falling and the lines a curve approaches
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsGraphs of the inverse functions

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