Question

solve for c. do not convert your answer to decimal form. rationalize the denominator and simplify your answer if possible. draw (image of a right triangle with one angle 30 degrees, one leg 7, the other leg c, and the right angle at the bottom left)

Explanation:

Step1: Identify trigonometric ratio

In a right triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 30^\circ\), opposite side to \(30^\circ\) is \(c\), adjacent side is \(7\). So \(\tan(30^\circ)=\frac{c}{7}\).

Step2: Recall \(\tan(30^\circ)\) value

We know that \(\tan(30^\circ)=\frac{1}{\sqrt{3}}\). Substitute into the equation: \(\frac{1}{\sqrt{3}}=\frac{c}{7}\).

Step3: Solve for \(c\)

Cross - multiply to get \(c = \frac{7}{\sqrt{3}}\). Now rationalize the denominator by multiplying numerator and denominator by \(\sqrt{3}\): \(c=\frac{7\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{7\sqrt{3}}{3}\).

Answer:

\(\frac{7\sqrt{3}}{3}\)