Question

find the measure of the side of the right triangle whose length is designated by a lowercase letter.
a ≈ □ cm
(round to the nearest whole number as needed.)

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\), \(\angle C = 90^\circ\), \(\angle A = 39^\circ\), and \(b = 320\) cm (adjacent to \(\angle A\)), \(a\) is the opposite side to \(\angle A\). We use the tangent function: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(39^\circ)=\frac{a}{b}\).

Step2: Solve for \(a\)

Substitute \(b = 320\) and \(\theta = 39^\circ\) into the formula: \(a = b\times\tan(39^\circ)\). Calculate \(\tan(39^\circ)\approx0.8098\), then \(a = 320\times0.8098\approx259.136\).

Step3: Round to the nearest whole number

Rounding \(259.136\) to the nearest whole number gives \(259\).

Answer:

\(259\)