The problem asks us to express $tan(\theta)$ in terms of $x$, given a right triangle with legs of length $x$ and $y$, hypotenuse of length $\sqrt{52}$, and $\theta$ as the angle opposite the side with length $x$.

1. Problem Description

The problem asks us to express $tan(\theta)$ in terms of $x$, given a right triangle with legs of length $x$ and $y$, hypotenuse of length $\sqrt{52}$, and $\theta$ as the angle opposite the side with length $x$.

2. Solution Steps

We have a right triangle with legs $x$ and $y$, and hypotenuse $\sqrt{52}$. We want to express $\tan(\theta)$ in terms of $x$. By definition,

$tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{y}$.

We need to express $y$ in terms of $x$. Since this is a right triangle, we can use the Pythagorean theorem:

$x^2 + y^2 = (\sqrt{52})^2$

$x^2 + y^2 = 52$

$y^2 = 52 - x^2$

$y = \sqrt{52 - x^2}$

Now we substitute this expression for $y$ into the tangent equation:

$tan(\theta) = \frac{x}{\sqrt{52 - x^2}}$.

3. Final Answer

$tan(\theta) = \frac{x}{\sqrt{52 - x^2}}$