Question

solve the right triangle.
round your answers to the nearest tenth.
a =
a =
b =

Explanation:

Step1: Find angle \(A\)

In a right - triangle, the sum of angles is \(180^{\circ}\). Since one angle is \(90^{\circ}\) and another is \(47^{\circ}\), then \(A = 90^{\circ}-47^{\circ}\).
\(A = 43^{\circ}\)

Step2: Find side \(a\)

Using the sine function \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here \(\theta = 47^{\circ}\), hypotenuse \(c = 24\), and opposite side is \(a\).
\(\sin(47^{\circ})=\frac{a}{24}\)
\(a = 24\times\sin(47^{\circ})\)
Using a calculator, \(\sin(47^{\circ})\approx0.7314\)
\(a\approx24\times0.7314 = 17.6\)

Step3: Find side \(b\)

Using the cosine function \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 47^{\circ}\), hypotenuse \(c = 24\), and adjacent side is \(b\).
\(\cos(47^{\circ})=\frac{b}{24}\)
\(b = 24\times\cos(47^{\circ})\)
Using a calculator, \(\cos(47^{\circ})\approx0.6820\)
\(b\approx24\times0.6820=16.4\)

Answer:

\(A = 43^{\circ}\), \(a\approx17.6\), \(b\approx16.4\)