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Chapter 5: Problem 17

Evaluate \(\tan \left(\cos ^{-1} \frac{1}{3}\right)\)

Short Answer

Expert verified
\(\tan \left(\cos ^{-1} \frac{1}{3}\right) = \sqrt{8}\)

Step by step solution

01

Compute the angle with cosine equal to \(\frac{1}{3}\)

Let's denote the angle whose cosine is \(\frac{1}{3}\) by \(\theta\). Then, we have: \[\cos{\theta} = \frac{1}{3}\] Now, let's move forward to find the sine of the angle \(\theta\).
02

Compute the sine of the angle

We can use the Pythagorean identity to relate sine and cosine: \[\sin^2{\theta} + \cos^2{\theta} = 1\] We know the value of the cosine from step 1: \[\sin^2{\theta} + \left(\frac{1}{3} \right)^2 = 1\] Now we solve for \(\sin{\theta}\): \[\sin^2{\theta} = 1 - \left(\frac{1}{3} \right)^2\] \[\sin{\theta} = \pm\sqrt{1 - \left(\frac{1}{3} \right)^2}\] Since the range of the inverse cosine function is in \([0, \pi]\), the angle \(\theta\) is in the first or second quadrant. Therefore, its sine is non-negative, and we choose the positive square root: \[\sin{\theta} = \sqrt{1 - \left(\frac{1}{3} \right)^2}\]
03

Compute the tangent of the angle

Now that we have both the sine and cosine of angle \(\theta\), we can find the tangent: \[\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\] Substitute the given values for sine and cosine: \[\tan{\theta} = \frac{\sqrt{1 - \left(\frac{1}{3} \right)^2}}{\frac{1}{3}}\]
04

Simplify the expression

We can simplify the expression as follows: \[\tan{\theta} = \frac{\sqrt{1 - \frac{1}{9}}}{\frac{1}{3}}\] \[\tan{\theta} = \frac{\sqrt{\frac{8}{9}}}{\frac{1}{3}}\] \[\tan{\theta} = 3\sqrt{\frac{8}{9}}\] \[\tan{\theta} = \sqrt{8}\] So, the final answer is: \[\tan \left(\cos ^{-1} \frac{1}{3}\right) = \sqrt{8}\]

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Most popular questions from this chapter

Show that $$ \cos (\pi-\theta)=-\cos \theta $$ for every angle \(\theta\).

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with \(\tan u=-2\) and \(\tan v=-3\) Find exact expressions for the indicated quantities. $$ \cos (-v) $$

Is tangent an even function, an odd function, or neither?

Is arcsine an even function, an odd function, or neither?

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with \(\tan u=-2\) and \(\tan v=-3\) Find exact expressions for the indicated quantities. $$ \tan (-v) $$
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