Question

a right triangle has a hypotenuse of length 3 inches. if one angle is 42°, find the length of each leg

the length of the shorter leg is □ inches, and the longer leg is □ inches.
(do not round until the final answer. then round to the nearest tenth as needed)

Explanation:

Step1: Use sine and cosine functions

In a right - triangle, if the hypotenuse \(c = 3\) inches and one angle \(\theta=42^{\circ}\).
The shorter leg (opposite to the smaller non - right angle) is \(a = c\sin\theta\), and the longer leg (adjacent to the smaller non - right angle) is \(b = c\cos\theta\).
Since \(\sin42^{\circ}\approx0.6691\) and \(\cos42^{\circ}\approx0.7431\)

Step2: Calculate the length of the shorter leg

The shorter leg \(a = 3\times\sin42^{\circ}\)
\(a = 3\times0.6691=2.0073\)

Step3: Calculate the length of the longer leg

The longer leg \(b = 3\times\cos42^{\circ}\)
\(b = 3\times0.7431 = 2.2293\)

Answer:

The length of the shorter leg is \(2.0\) inches, and the longer leg is \(2.2\) inches.