Question

solve for ( a ). do not convert your answer to decimal form. rationalize the denominator and simplify your answer if possible.
draw
(right triangle with one leg 10, one leg ( a ), right angle, and the angle opposite leg 10 is ( 60^circ ))
show your work here
hint: to add the square root symbol (( sqrt{} )), type
oot\

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, we know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 60^{\circ}\), the opposite side to \(\theta\) is \(10\) and the adjacent side is \(a\). So, \(\tan(60^{\circ})=\frac{10}{a}\).
We know that \(\tan(60^{\circ})=\sqrt{3}\). So the equation becomes \(\sqrt{3}=\frac{10}{a}\).

Step2: Solve for \(a\)

To solve for \(a\), we can cross - multiply. From \(\sqrt{3}=\frac{10}{a}\), we get \(a\times\sqrt{3}=10\). Then, \(a = \frac{10}{\sqrt{3}}\).

Step3: Rationalize the denominator

To rationalize the denominator of \(\frac{10}{\sqrt{3}}\), we multiply the numerator and the denominator by \(\sqrt{3}\). So, \(a=\frac{10\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{10\sqrt{3}}{3}\).

Answer:

\(\frac{10\sqrt{3}}{3}\)