/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Evaluate \(\tan \left(\sin ^{-1}... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 18

Evaluate \(\tan \left(\sin ^{-1} \frac{2}{5}\right)\)

Short Answer

Expert verified
The short answer to evaluating \(\tan \left(\sin ^{-1} \frac{2}{5}\right)\) is \(\frac{2}{\sqrt{21}}\).

Step by step solution

01

Identify the angle 胃

Let \(\theta = \sin ^{-1} \frac{2}{5}\). Then, \(\sin{\theta} = \frac{2}{5}\).
02

Find the cosine of the angle 胃

Now, we can create a right-angled triangle where one of the angles is 胃, with the opposite side having length 2 and the hypotenuse having length 5. We can use the Pythagorean theorem to find the length of the adjacent side (let's call it "a"): \(a^2 + 2^2 = 5^2\) \(a^2 = 25-4\) \(a^2 = 21\) \(a = \sqrt{21}\) The cosine of the angle 胃 is the ratio of the adjacent side to the hypotenuse, so: \(\cos{\theta} = \frac{\sqrt{21}}{5}\)
03

Find the tangent of the angle 胃

Finally, we use the identity \(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\): \(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{\frac{2}{5}}{\frac{\sqrt{21}}{5}} = \frac{2}{\sqrt{21}}\) Thus, \(\tan \left(\sin ^{-1} \frac{2}{5}\right) = \frac{2}{\sqrt{21}}\).

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