Problem 895 Evaluate (a) \(\sin ^{-1}(\sqrt{... [FREE SOLUTION]

Chapter 28: Problem 895

Evaluate (a) \(\sin ^{-1}(\sqrt{3} / 2)\), (b) \(\tan ^{-1}(-\sqrt{3})\).

Short Answer

Expert verified

(a) \(\sin ^{-1}(\sqrt{3} / 2) = 60^{\circ}\) (b) \(\tan ^{-1}(-\sqrt{3}) = -60^{\circ}\)

Step by step solution

Recall the properties of the inverse sine function.

The principal range of the inverse sine function is [-90掳, 90掳]. Therefore, we are looking for an angle within this range whose sine value is \(\sqrt{3} / 2\).

Find the angle with sine value of \(\sqrt{3} / 2\).

By recalling the standard angles where sine values are well known, we can deduce that at a 60-degree angle, the sine value is \(\sqrt{3} / 2\): \[\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\]

Evaluate the inverse sine function.

Now, we can compute the inverse sine of \(\sqrt{3}/2\): \[\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = 60^{\circ}\] (b) Evaluate \(\tan ^{-1}(-\sqrt{3})\).

Recall the properties of the inverse tangent function.

The principal range of the inverse tangent function is (-90掳, 90掳). Therefore, we are looking for an angle within this range whose tangent value is \(-\sqrt{3}\).

Find the angle with tangent value of \(-\sqrt{3}\).

By recalling the standard angles where tangent values are well known, we can deduce that at a 60-degree angle, the tangent value is \(\sqrt{3}\): \[\tan(60^{\circ}) = \sqrt{3}\] However, since we are looking for a negative value of \(\sqrt{3}\), we need to consider the negative angle -60 degrees. Consequently, in the principal range of inverse tangent, we have: \[\tan(-60^{\circ}) = -\sqrt{3}\]

Evaluate the inverse tangent function.

Now, we can compute the inverse tangent of \(-\sqrt{3}\): \[\tan^{-1}\left(-\sqrt{3}\right) = -60^{\circ}\]

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91影视

Evaluate (a) \(\sin ^{-1}(\sqrt{3} / 2)\), (b) \(\tan ^{-1}(-\sqrt{3})\).

Short Answer

Step by step solution

Recall the properties of the inverse sine function.

Find the angle with sine value of \(\sqrt{3} / 2\).

Evaluate the inverse sine function.

Recall the properties of the inverse tangent function.

Find the angle with tangent value of \(-\sqrt{3}\).

Evaluate the inverse tangent function.

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