Question 11 of 20
This question: 1 point(s) possible
Submite

Solve the right triangle ABC , with .

(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)
ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)
ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

Ask by Ingram Ford. in the United States

Mar 13,2025

Question

Question 11 of 20
This question: 1 point(s) possible
Submite

Solve the right triangle ABC , with .

(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)
ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)
ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

Ask by Ingram Ford. in the United States

Mar 13,2025

UpStudy AI · Accepted Answer

We have a right triangle with angle $$ C = 90^\circ $$, angle $$ A = 57.4^\circ $$, and the hypotenuse $$ c = 22.4 $$ ft. The goal is to find angle $$ B $$, side $$ a $$ (opposite angle $$ A $$), and side $$ b $$ (opposite angle $$ B $$).

1. Since the sum of the angles in a triangle is $$ 180^\circ $$ and $$ C=90^\circ $$, we have:
   $$
   A + B + C = 180^\circ \quad \Longrightarrow \quad 57.4^\circ + B + 90^\circ = 180^\circ.
   $$
   Solving for $$ B $$:
   $$
   B = 180^\circ - 90^\circ - 57.4^\circ = 32.6^\circ.
   $$

2. Use trigonometric ratios to find the missing sides. Since $$ c $$ is the hypotenuse, we use:
   - For side $$ a $$ (opposite angle $$ A $$):
     $$
     \sin A = \frac{a}{c} \quad \Longrightarrow \quad a = c \sin A.
     $$
     Plug in the known values:
     $$
     a = 22.4 \sin(57.4^\circ).
     $$
     Calculating (using a calculator for the sine value):
     $$
     \sin 57.4^\circ \approx 0.8387 \quad \Longrightarrow \quad a \approx 22.4 \times 0.8387 \approx 18.8 \text{ ft}.
     $$

3. For side $$ b $$ (opposite angle $$ B $$):
   $$
   \sin B = \frac{b}{c} \quad \Longrightarrow \quad b = c \sin B.
   $$
   Use the known values:
   $$
   b = 22.4 \sin(32.6^\circ).
   $$
   Calculate:
   $$
   \sin 32.6^\circ \approx 0.5394 \quad \Longrightarrow \quad b \approx 22.4 \times 0.5394 \approx 12.1 \text{ ft}.
   $$

Thus, the solutions to the triangle are:
$$
B=32.6^\circ, \quad a=18.8 \text{ ft}, \quad b=12.1 \text{ ft}.
$$
$$ B = 32.6^\circ $$, $$ a = 18.8 $$ ft, $$ b = 12.1 $$ ft

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