Problem 23 - Precalculus Essentials

Question

Find the exact value of each expression. $$\tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

\(\tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right) = \sqrt{3}-2\)

Step 1: Recall the Tangent Subtraction Formula

The formula for the tangent of a difference of two angles \(A\) and \(B\) is \(\tan(A-B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}\). This formula will be used in the next step.

Step 2: Apply the Tangent Subtraction Formula

Applying the formula gives: \( \tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right) = \frac{\tan\left(\frac{4\pi}{3}\right) - \tan\left(\frac{\pi}{4}\right)}{1 + \tan\left(\frac{4\pi}{3}\right)\tan\left(\frac{\pi}{4}\right)}\)

Step 3: Evaluate Tangents and Simplify

From the unit circle, we know that: \( \tan\left(\frac{4\pi}{3}\right) = \sqrt{3} \) and \( \tan\left(\frac{\pi}{4}\right) = 1 \). Substituting these values in, we get: \( \tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right) = \frac{\sqrt{3} - 1}{1 + \sqrt{3}}\)

Step 4: Rationalize the Denominator

Multiplying the numerator and denominator by \(1-\sqrt{3}\) to rationalize the denominator gives: \( \tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right) = \frac{3 - 2\sqrt{3}}{-2}\)

Step 5: Simplify the Result

Finally, simplifying gives the exact value: \( \tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right) = \sqrt{3}-2\)