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Chapter 4: Problem 43

Find the exact value of each expression. Do not use a calculator. $$\frac{\tan \frac{\pi}{3}}{2}-\frac{1}{\sec \frac{\pi}{6}}$$

Short Answer

Expert verified
-3sqrt{3}/2

Step by step solution

01

Simplify the numerator

First, replace the \(\tan(\pi/3) \) with its known value. Based on the unit circle or right triangle definitions, the \(\tan(\pi/3) \) is \(\sqrt{3}. \) So the problem becomes \(\frac{\sqrt{3}}{2} - \frac{1}{\sec(\pi/6)} \)
02

Simplify the denominator

Next, simplify the term in the denominator, \( \sec(\pi/6) \). The secant function is the reciprocal of the cosine function, so \( \sec(\pi/6) = 1/\cos(\pi/6) \). The cosine of \(\pi/6\) is \(\sqrt{3}/2\), so substituting, the problem becomes \( \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{3}/2}\)
03

Simplify the subtraction

The next step is to simplify the fraction by multiplying the numerator and denominator by 2, which erases the fraction: \( \frac{\sqrt{3}}{2} - 2\sqrt{3} \), which finally simplifies to \( \frac{\sqrt{3}-4\sqrt{3}}{2} \). By simplifying the numerators we get \( -\frac{3\sqrt{3}}{2} \) as our result.

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Most popular questions from this chapter

Use a graphing utility to graph $$ y=\sin x+\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+\frac{\sin 4 x}{4} $$ in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by [-2,2,1] viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \cos (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=\sin x, g(x)=\cos 2 x, h(x)=(f-g)(x)$$

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Without drawing a graph, describe the behavior of the basic cosine curve.
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