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Chapter 4: Problem 71

Write an algebraic expression that is equivalent to the given expression. $$\tan \left(\arccos \frac{x}{3}\right)$$

Short Answer

Expert verified
The algebraic expression equivalent to \(\tan \left(\arccos \frac{x}{3}\right)\) is \(\frac{\sqrt{9 - x^2}}{x}\).

Step by step solution

01

Understand the Given Expression

The given expression is the tangent of the arccosine of \(x/3\). Arccosine is the inverse function of cosine. It undoes the action of the cosine function. The expression inside the parentheses, \(\arccos \frac{x}{3}\), represents the angle with a cosine of \(x/3\). The outer function, \(\tan\), then finds the tangent of this angle.
02

Apply the Definition of Tangent

Remember that tangent is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. We can use this to define the angle \(\arccos \frac{x}{3}\). After drawing an imaginary right triangle, one can define the side adjacent to the angle to be \(x\), whereas the hypotenuse is \(3\). By the definition of cosine, the cosine of an angle in a right triangle is \(\cos \theta = \frac{adjacent}{hypotenuse}\). Therefore, cosine of our angle is \(\cos \arccos \frac{x}{3} = \frac{x}{3}\). According to Pythagoras' law, we can find the length of the opposite side. It would be \(\sqrt{hypotenuse^2 - adjacent^2} = \sqrt{9 - x^2}\) assuming x is positive.
03

Substitute Values into the Tangent Function

The tangent of our angle would be the ratio of the opposite to the adjacent side. Substituting our values into the tangent formula gives: \(\tan \left(\arccos \frac{x}{3}\right) = \frac{\sqrt{9 - x^2}}{x}\).

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Most popular questions from this chapter

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{64} \sin 792 \pi t$$

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