Question

determine the unknown angles or sides of these right triangles. compute angles to the nearest minute in triangles with customary unit sides. compute angles to the nearest hundredth degree in triangles with metric unit sides. compute sides to 3 decimal places. 27. determine \\( \angle a \\).

Explanation:

Step1: Identify the sides relative to ∠A

In the right triangle, the side opposite to ∠A is \( 3.900 \) in, and the side adjacent to ∠A is \( 4.870 \) in. We use the tangent function, where \( \tan(A)=\frac{\text{opposite}}{\text{adjacent}} \).
\( \tan(A)=\frac{3.900}{4.870} \)

Step2: Calculate the arctangent

First, compute \( \frac{3.900}{4.870}\approx0.799 \). Then, find \( A = \arctan(0.799) \). Using a calculator, we get \( A\approx38.63^\circ \). Now, convert the decimal part to minutes. The decimal part is \( 0.63 \), and since \( 1^\circ = 60' \), we calculate \( 0.63\times60\approx38' \). So, \( A\approx38^\circ38' \).

Answer:

\( \angle A\approx38^\circ38' \) (or approximately \( 38.63^\circ \))