Math / Trigonometry

QuestionQuestion 20 of 23 This test: 23 point(s) possible This question: 1 point(s) possible
Solve the right triangle with $\mathrm{B}=55.4^{\circ}$ and $\mathrm{c}=27.2$ . Round off the results according to the table below. \begin{tabular}{cc} \hline \begin{tabular}{c} Measurements of \\ Angle to Nearest \end{tabular} & \begin{tabular}{c} Accuracy of \\ Trigonometric \\ Function \end{tabular} \\ \hline $1^{\circ}$ & 2 significant digits \\ $0.1^{\circ}$ & 3 significant digits \\ $0.01^{\circ}$ & 4 significant digits \\ \hline \end{tabular}
Select the correct answer below and, if necessary, fill in the answer boxes to complete your choice. A. $A=34.6{ }^{\circ}, \mathrm{a}=11.1, \mathrm{~b}=24.8$ B. There is not enough information to solve the triangle.

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. Angle $B = 55.4^\circ$ .
3. The hypotenuse $c = 27.2$ .
4. We need to find the missing angle $A$ , and the sides $a$ and $b$ .
5. The angle $A$ should be rounded to the nearest $0.1^\circ$ .
6. The sides $a$ and $b$ should be calculated to 3 significant digits.

STEP 2

1. Find the missing angle $A$ .
2. Use trigonometric functions to find side $a$ .
3. Use trigonometric functions to find side $b$ .
4. Round the results according to the given table.

STEP 3

Find the missing angle $A$ using the fact that the sum of angles in a triangle is $180^\circ$ . Since it is a right triangle:
$A = 90^\circ - B$ $A = 90^\circ - 55.4^\circ$ $A = 34.6^\circ$

STEP 4

Use the sine function to find side $a$ :
$\sin(B) = \frac{b}{c}$ $b = c \cdot \sin(B)$ $b = 27.2 \cdot \sin(55.4^\circ)$
Calculate $b$ :
$b \approx 27.2 \cdot 0.826$ $b \approx 22.4672$
Round $b$ to 3 significant digits:
$b \approx 22.5$

STEP 5

Use the cosine function to find side $a$ :
$\cos(B) = \frac{a}{c}$ $a = c \cdot \cos(B)$ $a = 27.2 \cdot \cos(55.4^\circ)$
Calculate $a$ :
$a \approx 27.2 \cdot 0.5736$ $a \approx 15.60672$
Round $a$ to 3 significant digits:
$a \approx 15.6$

STEP 6

Verify the results and compare with the given options. The calculated values are:
$A = 34.6^\circ, \, a = 15.6, \, b = 22.5$
These do not match option A exactly. Therefore, the correct answer is:
B. There is not enough information to solve the triangle.

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