Question

solve the right triangle. round your answers to the nearest tenth. image of a right triangle with right angle at the bottom left, labeled with: vertical leg (from right angle to top vertex b) length 29, horizontal leg (from right angle to bottom right vertex) labeled b, hypotenuse (from b to bottom right vertex) labeled c, and the bottom right angle is 47 degrees. to the right of the triangle, there are three boxes to fill: b = °, b = , c =

Explanation:

Step1: Find angle \( B \)

In a right triangle, the sum of the acute angles is \( 90^\circ \). Given one acute angle is \( 47^\circ \), so \( B = 90^\circ - 47^\circ = 43^\circ \).

Step2: Find side \( b \)

We know the opposite side to \( 47^\circ \) is \( 29 \) (let's call the right angle \( C \), so side opposite \( 47^\circ \) is \( BC = 29 \), adjacent side is \( b \)). Using tangent: \( \tan(47^\circ) = \frac{29}{b} \), so \( b = \frac{29}{\tan(47^\circ)} \). Calculate \( \tan(47^\circ) \approx 1.0724 \), then \( b \approx \frac{29}{1.0724} \approx 27.0 \).

Step3: Find side \( c \) (hypotenuse)

Using sine: \( \sin(47^\circ) = \frac{29}{c} \), so \( c = \frac{29}{\sin(47^\circ)} \). Calculate \( \sin(47^\circ) \approx 0.7314 \), then \( c \approx \frac{29}{0.7314} \approx 39.6 \).

Answer:

\( B = 43^\circ \)
\( b \approx 27.0 \)
\( c \approx 39.6 \)