Problem 43 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

-3sqrt{3}/2

Step 1: 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 \(\frac{\sqrt{3}}{2} - \frac{1}{\sec(\pi/6)} \)

Step 2: 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 \( \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{3}/2}\)

Step 3: 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.