Which of the following statements are true? I. sin^-1(sin x) = x for every real x. II. sin(sin^-1 x) = x for every x with -1 ≤ x ≤ 1. III. cos^-1(cos x) = x for every x with -π/2 ≤ x ≤ π/2. IV. tan(tan^-1 x) = x for every real x.

Question 101

MediumOriginal question

Which of the following statements are true?
I. $\sin^{-1}(\sin x) = x$ for every real $x$ .
II. $\sin\left(\sin^{-1} x\right) = x$ for every $x$ with $-1 \le x \le 1$ .
III. $\cos^{-1}(\cos x) = x$ for every $x$ with $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$ .
IV. $\tan\left(\tan^{-1} x\right) = x$ for every real $x$ .

II and IV onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerI, II and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

A function applied to its own inverse gives back $x$ wherever the inverse is defined: II and IV are true.
The other order gives back $x$ only on the branch: $\sin^{-1}(\sin \pi) = 0$ , so I is false; $\cos^{-1}\left(\cos\left(-\frac{\pi}{4}\right)\right) = \frac{\pi}{4}$ , so III is false.

Worth noting

III uses the branch of $\sin^{-1}$ ; $\cos^{-1}$ returns only angles in $[0, \pi]$ , never a negative one.

Concept: When an inverse undoes a function
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsValues of expressions

Practise this chapter →

Ten questions from Inverse Trigonometric Functions, at your own pace.

Explanation