Solve the right triangle ABC , with $$ \mathrm{C}=90^{\circ} $$. $$ \mathrm{A}=59.5^{\circ}, \mathrm{c}=21.2 \mathrm{ft} $$ $$ \mathrm{B}=\square^{\circ} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.) $$ \mathrm{a}=\square_{\mathrm{ft}} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.) $$ \mathrm{b}=\square \mathrm{ft} $$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

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UpStudy AI · Accepted Answer

To solve the right triangle $$ ABC $$ with $$ \angle C = 90^\circ $$, $$ \angle A = 59.5^\circ $$, and side $$ c = 21.2 $$ ft, follow these steps:

1. **Find Angle $$ B $$:**

Since the sum of angles in a triangle is $$ 180^\circ $$:
   $$
   \angle B = 180^\circ - \angle A - \angle C = 180^\circ - 59.5^\circ - 90^\circ = 30.5^\circ
   $$

2. **Find Side $$ a $$ (opposite $$ \angle A $$):**

Use the sine function:
   $$
   \sin(\angle A) = \frac{a}{c} \implies a = c \cdot \sin(\angle A)
   $$
   $$
   \sin(59.5^\circ) \approx 0.8616
   $$
   $$
   a = 21.2 \, \text{ft} \times 0.8616 \approx 18.3 \, \text{ft}
   $$

3. **Find Side $$ b $$ (opposite $$ \angle B $$):**

Similarly, use the sine function:
   $$
   \sin(\angle B) = \frac{b}{c} \implies b = c \cdot \sin(\angle B)
   $$
   $$
   \sin(30.5^\circ) \approx 0.5075
   $$
   $$
   b = 21.2 \, \text{ft} \times 0.5075 \approx 10.8 \, \text{ft}
   $$

**Summary of Results:**
- $$ \angle B = 30.5^\circ $$
- $$ a = 18.3 $$ ft
- $$ b = 10.8 $$ ft
$$ \angle B = 30.5^\circ $$, $$ a = 18.3 $$ ft, $$ b = 10.8 $$ ft

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