Problem 21 - Precalculus Essentials

Question

Find the exact value of each expression. $$\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The exact value of tan((pi/6)+(pi/4)) is \( \sqrt{3} \)

Step 1: Identify the Trigonometric Identity

The expression \(\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\right)\) is the sum of two angles. The tangent of the sum of two angles can be expressed using the formula: \(\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \cdot \tan(\beta)}\). Here, \(\alpha = \frac{\pi}{6}\) and \(\beta = \frac{\pi}{4}\)

Step 2: Find \(\tan(\alpha)\) and \(\tan(\beta)\)

Now that we know the formula to use, we need to find the tangent of the individual angles. The tangent of \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\) are \(\frac{\sqrt{3}}{3}\) and \(1\) respectively.

Step 3: Substitute into the Identity

We substitute the values obtained into the formula: \(\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \cdot \tan(\beta)} = \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}\)

Step 4: Simplify the Expression

Carry out the arithmetic in the numerator and the denominator and simplify the expression. This results in \( \frac{1 + \sqrt{3}/3}{1 - \sqrt{3}/3}\) which simplifies further to reach the exact value \( \sqrt{3} \).