Question
Find \( c \), the hypotenuse, rounded to four decimal places, in a right triangle with \( b=16 \) and \( B \) \( =55^{\circ} \). (1 point) 11.2033 o 19.5324 o 22.6274 27.8951
Ask by Carter Willis. in the United States
Jan 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The hypotenuse \( c \) is approximately 19.5324.
Solution
To find the hypotenuse \( c \) in a right triangle with side \( b = 16 \) and angle \( B = 55^\circ \):
1. **Use the sine function:**
\[
\sin(B) = \frac{b}{c}
\]
2. **Rearrange to solve for \( c \):**
\[
c = \frac{b}{\sin(B)}
\]
3. **Calculate \( \sin(55^\circ) \):**
\[
\sin(55^\circ) \approx 0.8192
\]
4. **Substitute the values:**
\[
c = \frac{16}{0.8192} \approx 19.5324
\]
**Answer:** 19.5324
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Simplify this solution Extra Insights
To find \( c \), the hypotenuse of the right triangle, we can use the trigonometric relationship involving the sine function. Since we have one side (the adjacent side, \( b \)) and the angle \( B \), we can use the formula: \[ c = \frac{b}{\cos(B)} \] First, let's plug in the values: \[ c = \frac{16}{\cos(55^{\circ})} \] Calculating \( \cos(55^{\circ}) \) gives us approximately 0.5736. Now, we can compute: \[ c \approx \frac{16}{0.5736} \approx 27.8951 \] Thus, \( c \) rounded to four decimal places is **27.8951.**
