State the values between which each of the following must lie

(a)
the principal value of $\tan^{-1} x$ .
(b)
the principal value of $\cos^{-1} 2x$ .

It won't just hand you the answer — it asks where you're stuck.How the checker works

Worked solution

1(a)

1. State the principal value range for the inverse tangent function

The principal value of $\tan^{-1} x$ lies strictly between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ radians (or $-90^\circ$ and $90^\circ$ ).
$-\frac{\pi}{2} < \tan^{-1} x < \frac{\pi}{2}$

Answer

$-\frac{\pi}{2} < \tan^{-1} x < \frac{\pi}{2}$

1(b)

1. State the range of the principal value for the inverse cosine function

The principal value of $\cos^{-1}(\theta)$ is defined to lie in the interval $[0, \pi]$ . The argument $2x$ affects the domain of $x$ , but the range of the principal value remains the same.
$0 \le \cos^{-1} 2x \le \pi$

Answer

$0$ and $\pi$ (or $0^\circ$ and $180^\circ$ )

AI analysis — may contain mistakes.

Worked this one out on paper? Snap your steps and Gigle's AI math checker will circle the step where it went wrong.

State the values between which each of the following must lie

Worked solution

1(a)

1(b)

More questions on this topic