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Chapter 5: Problem 16

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

Short Answer

Expert verified
The short answer based on the step-by-step solution is: \(\cos(\sin^{-1}\frac{2}{5}) = \frac{\sqrt{21}}{5}\).

Step by step solution

01

Draw a right triangle

Draw a right triangle such that one of the angles, \(\theta\), satisfies \(\sin\theta = \frac{2}{5}\). Recall that the sine of an angle is the ratio of the opposite side to the hypotenuse. In our case, let the side opposite to the angle \(\theta\) have length 2 and the hypotenuse have length 5.
02

Use the Pythagorean theorem to find the adjacent side

To determine the length of the side adjacent to the angle \(\theta\), we'll use the Pythagorean theorem: \[a^2 + b^2 = c^2\] where \(a\) and \(b\) are the side lengths of the two legs of the right triangle, and \(c\) is the length of the hypotenuse. In our triangle, we can substitute the known values: \[a^2 + 2^2 = 5^2\] \[a^2 + 4 = 25\] \[a^2 = 21\] \[a = \sqrt{21}\] So, the side adjacent to the angle \(\theta\) has length \(\sqrt{21}\).
03

Evaluate the given expression

Now that we know the side lengths of the right triangle, we can find the value of \(\cos(\sin^{-1}\frac{2}{5})\). Recall that the cosine of an angle is the ratio of the adjacent side to the hypotenuse: \[\cos(\sin^{-1}\frac{2}{5}) = \cos(\theta) = \frac{\sqrt{21}}{5}\] The desired result is \(\cos(\sin^{-1} \frac{2}{5}) = \frac{\sqrt{21}}{5}\).

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