Given a right triangle with an angle of $30°$ and opposite side $a$ and adjacent side $b$, which equation can be used to solve for $b$? $\tan (30°)$ =

• A. $\frac{a}{\sqrt{3}}$
• B. $\frac{b}{a}$
• C. $\frac{a}{b}$
• D. $\frac{b}{\sqrt{3}}$

Answer: (C)

Verified step by step guidance

1. Recall the definition of the tangent function in a right triangle: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
2. Identify the given angle \(30^\circ\), the opposite side \(a\), and the adjacent side \(b\) in the triangle.
3. Write the tangent ratio for the \(30^\circ\) angle: \(\tan(30^\circ) = \frac{a}{b}\).
4. Use the known exact value of \(\tan(30^\circ)\), which is \(\frac{1}{\sqrt{3}}\), to set up the equation: \(\frac{1}{\sqrt{3}} = \frac{a}{b}\).
5. Solve this equation for \(b\) by cross-multiplying to get \(b = a \sqrt{3}\), which is the equation to find \(b\) in terms of \(a\).