Question

solve for x. round to the nearest tenth, if necessary.

Explanation:

Step1: Identify trigonometric ratio

In right - triangle $LMN$ with right - angle at $M$, we know the side adjacent to $\angle L$ is $ML = 80$ and we want to find the hypotenuse $LN=x$. We use the cosine function since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 59^{\circ}$, adjacent side to $\theta$ is $ML$ and hypotenuse is $LN$. So, $\cos(59^{\circ})=\frac{ML}{LN}$.

Step2: Substitute values and solve for $x$

We know that $\cos(59^{\circ})\approx0.515$ and $ML = 80$, $LN=x$. Substituting into the formula $\cos(59^{\circ})=\frac{80}{x}$, we can cross - multiply to get $x\cos(59^{\circ})=80$. Then $x=\frac{80}{\cos(59^{\circ})}$.

Step3: Calculate the value of $x$

$x=\frac{80}{0.515}\approx155.3$.

Answer:

$x\approx155.3$