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.)
b = 70 cm

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \(ABC\) with right - angle at \(C\), we know the adjacent side \(b = 70\) cm to angle \(A=35^{\circ}\) and we want to find the opposite side \(a\) to angle \(A\). We use the tangent function since \(\tan A=\frac{\text{opposite}}{\text{adjacent}}\).
\(\tan A=\frac{a}{b}\)

Step2: Substitute the known values

We know that \(A = 35^{\circ}\) and \(b = 70\) cm. Substituting into the formula \(\tan A=\frac{a}{b}\), we get \(\tan35^{\circ}=\frac{a}{70}\).

Step3: Solve for \(a\)

Multiply both sides of the equation by \(70\): \(a = 70\times\tan35^{\circ}\).
Since \(\tan35^{\circ}\approx0.7002\), then \(a = 70\times0.7002=49.014\) cm.

Step4: Round the result

Rounding \(49.014\) to the nearest whole number, we get \(a\approx49\) cm.

Answer:

49