Question

solve for the variable shown.
y = \boxed{} (round to three decimal places as needed.)

Explanation:

Step1: Identify trigonometric ratio

The triangle is right - angled (implied by the right angle in the diagram). We have the angle \(48^{\circ}\), the adjacent side to the angle is \(y\) and the opposite side is \(42\). The trigonometric ratio that relates the opposite and adjacent sides of an angle in a right - triangle is the tangent function. The formula for tangent is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 48^{\circ}\), opposite \(= 42\) and adjacent \(= y\). So, \(\tan(48^{\circ})=\frac{42}{y}\).

Step2: Solve for y

We can re - arrange the formula \(\tan(48^{\circ})=\frac{42}{y}\) to solve for \(y\). Cross - multiplying gives us \(y\times\tan(48^{\circ}) = 42\), then \(y=\frac{42}{\tan(48^{\circ})}\).
We know that \(\tan(48^{\circ})\approx1.1106125148\).
Substitute this value into the formula for \(y\): \(y=\frac{42}{1.1106125148}\approx37.816\) (rounded to three decimal places).

Answer:

\(37.816\)