Solve the right triangle​ ABC, with Cequals90degrees.
Aequals34degrees 15prime​, cequals618.5 ft
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Part 1
Bequals
  
enter your response heredegrees 
  
enter your response hereprime ​(Simplify your answers. Type an integer or a decimal.​)
aequals
  
enter your response here ft ​(Simplify your answers. Type an integer or a decimal. Round to the nearest tenth if​ needed.)
bequals
  
enter your response here ft ​(Simplify your answers. Type an integer or a decimal. Round to the nearest tenth if​ needed.)

To solve the right triangle \(ABC\) where \(C = 90^\circ\), \(A = 34^\circ 15'\), and \(c = 618.5\) ft, we will:

1. Find angle \(B\),
2. Find side \(a\) (opposite to angle \(A\)),
3. Find side \(b\) (opposite to angle \(B\)).

### Step 1: Find angle \(B\)
Since the sum of angles in any triangle is \(180^\circ\), and one angle is \(90^\circ\):

\[
B = 90^\circ - A
\]
Convert angle \(A = 34^\circ 15'\) to decimal form:
\[
34^\circ 15' = 34 + \frac{15}{60} = 34.25^\circ
\]
Now subtract from \(90^\circ\):
\[
B = 90^\circ - 34.25^\circ = 55.75^\circ
\]
Thus, \(B = 55^\circ 45'\) (since \(0.75^\circ = 45'\)).

### Step 2: Find side \(a\) (opposite angle \(A\))
Using the sine function:
\[
\sin(A) = \frac{a}{c}
\]
Solving for \(a\):
\[
a = c \cdot \sin(A) = 618.5 \cdot \sin(34.25^\circ)
\]
Calculating \( \sin(34.25^\circ) \):
\[
a \approx 618.5 \cdot 0.5620 \approx 347.5 \, \text{ft}
\]

### Step 3: Find side \(b\) (opposite angle \(B\))
Using the cosine function:
\[
\cos(A) = \frac{b}{c}
\]
Solving for \(b\):
\[
b = c \cdot \cos(A) = 618.5 \cdot \cos(34.25^\circ)
\]
Calculating \( \cos(34.25^\circ) \):
\[
b \approx 618.5 \cdot 0.8271 \approx 511.5 \, \text{ft}
\]

### Final Answers:
- \(B = 55^\circ 45'\)
- \(a \approx 347.5 \, \text{ft}\)
- \(b \approx 511.5 \, \text{ft}\)

Would you like further details or clarification?

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Here are 5 related questions:
1. How do you convert angles from degrees and minutes to decimal form?
2. What is the formula for finding the hypotenuse in a right triangle?
3. How does the sine function relate to right triangles?
4. Can you derive side lengths using other trigonometric functions?
5. How do you check if your triangle calculations are correct?

**Tip:** Always double-check the angle conversions to avoid errors when solving triangles.

Solve the right triangle ABC, with C = 90°, A = 34° 15', and c = 618.5 ft. Find B, a, and b (simplify answers, type an integer or decimal, round to the nearest tenth if needed).

Solve the right triangle ABC, where angle A = 34° 15', angle C = 90°, and side c = 618.5 ft. Learn how to calculate angle B, side a, and side b using trigonometric functions like sine and cosine. Detailed step-by-step solution provided, suitable for students in Grades 9-12.

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