State the values between which the principal value of sin⁡−1x\sin^{-1} x must lie. Find the principal value of tan⁡−11\tan^{-1} 1 in radian in exact form.

Principal valuesSpecial angles中文
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Worked solution

6(a)

1. State the range of the principal value of inverse sine

The principal value of sin⁡−1x\sin^{-1} x lies between −π2-\frac{\pi}{2} and π2\frac{\pi}{2} inclusive.

Answer

−π2⩽sin⁡−1x⩽π2-\frac{\pi}{2} \leqslant \sin^{-1} x \leqslant \frac{\pi}{2} (or −90∘⩽sin⁡−1x⩽90∘-90^\circ \leqslant \sin^{-1} x \leqslant 90^\circ)

6(b)

1. Evaluate the principal value of $\tan^{-1} 1$ in radians

Let θ=tan⁡−11\theta = \tan^{-1} 1.
Then tan⁡θ=1\tan \theta = 1.
Since the principal value of tan⁡−1x\tan^{-1} x lies in (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right), we have:
θ=π4\theta = \frac{\pi}{4}

Answer

π4\frac{\pi}{4}

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