Problem 24 - Precalculus Essentials

Question

Find the exact value of each expression. $$\tan \left(\frac{5 \pi}{3}-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The solution to the expression \( \tan \left(\frac{5 \pi}{3}-\frac{\pi}{4}\right) \) is \( -2 + \sqrt{3} \).

Step 1: Rewrite expression with angle subtraction formula for tangent

The expression \( \tan \left(\frac{5 \pi}{3}-\frac{\pi}{4}\right) \) can be rewritten using the formula from angle subtraction which results into \( \frac{\tan\left(\frac{5\pi}{3}\right) - \tan\left(\frac{\pi}{4}\right)}{1 + \tan\left(\frac{5\pi}{3}\right) \cdot \tan\left(\frac{\pi}{4}\right)} \).

Step 2: Substitute with known values

We can then substitute the known exact values of the trigonometric functions, which gives us: \( \frac{\tan\left(\frac{5\pi}{3}\right) - \tan\left(\frac{\pi}{4}\right)}{1 + \tan\left(\frac{5\pi}{3}\right) \cdot \tan\left(\frac{\pi}{4}\right)} = \frac{\sqrt{3} - 1}{1 + \sqrt{3} \cdot 1} = \frac{\sqrt{3} - 1}{1+\sqrt{3}} \).

Step 3: Simplify the expression

Next, we simplify the expression by multiplying numerator and denominator by the conjugate of the denominator which results in \( \frac{(\sqrt{3} - 1)(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})} \). This simplifies further to \( \frac{3 -2\sqrt{3} +1}{1-3} = - \frac{4 - 2\sqrt{3}}{2} = -2 + \sqrt{3} \).