Problem 22 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact value of the expression \(\tan \left(\frac{\pi}{3}+\frac{\pi}{4}\right)\) is \(-1 - \sqrt{3}\)

Step 1: Recognize the formula

Note that we have a trigonometric function that involves the sum of two angles, so we should use the formula for \(\tan(a+b)\), which is \(\frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\)

Step 2: Apply the formula to our specific problem

Let's replace \(a\) with \(\frac{\pi}{3}\), and \(b\) with \(\frac{\pi}{4}\) in our formula. This gives us \(\frac{\tan(\frac{\pi}{3}) + \tan(\frac{\pi}{4})}{1 - \tan(\frac{\pi}{3})\tan(\frac{\pi}{4})}\)

Step 3: Replace with known values of tan

We know that \(\tan(\frac{\pi}{3}) = \sqrt{3}\) and \(\tan(\frac{\pi}{4}) = 1\). Plugging these values into the equation we derived in Step 2, we get \(\frac{\sqrt{3} + 1}{1-\sqrt{3} \cdot 1}\).

Step 4: Simplify the expression

After performing the operations, we can simplify the expression to \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}} = -1 - \sqrt{3}\)