Question

there are two possible triangles with the
measures given. complete the steps to solve the
triangles.
b = 18.7, c = 16.1, c = 35°
which equation can you use to find the measure
of angle b?
\\( \frac { 16.1 } { \sin ( 35 ^ { \circ } ) } = \frac { 18.7 } { \sin ( b ) } \\)
\\( \frac { 18.7 } { \sin ( 35 ^ { \circ } ) } = \frac { 16.1 } { \sin ( b ) } \\)
\\( \frac { 18.7 } { 16.1 } = \frac { \sin ( 35 ^ { \circ } ) } { \sin ( b ) } \\)
from least to greatest, what are the possible
measures of angle b? round to the nearest whole
number.
\\( 42 ^ { \circ } \\) and \\( 138 ^ { \circ } \\)
complete
the unknown measures of one triangle are about
\\( a = 138 ^ { \circ }, b = 42 ^ { \circ }, a = 14.0 \\)
\\( a = 103 ^ { \circ }, b = 42 ^ { \circ }, a = 27.4 \\)
\\( a = 103 ^ { \circ }, b = 42 ^ { \circ }, a = 21.8 \\)
done

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\). Given \(b = 18.7\), \(c=16.1\), and \(C = 35^{\circ}\), we use the formula \(\frac{c}{\sin(C)}=\frac{b}{\sin(B)}\). Substituting the values, we get \(\frac{16.1}{\sin(35^{\circ})}=\frac{18.7}{\sin(B)}\).

Step2: Solve for \(\sin(B)\)

Cross - multiply the equation \(\frac{16.1}{\sin(35^{\circ})}=\frac{18.7}{\sin(B)}\) to obtain \(16.1\sin(B)=18.7\sin(35^{\circ})\). Then \(\sin(B)=\frac{18.7\sin(35^{\circ})}{16.1}\). Calculate \(\sin(35^{\circ})\approx0.5736\), so \(\sin(B)=\frac{18.7\times0.5736}{16.1}\approx\frac{10.726}{16.1}\approx0.666\).

Step3: Find the two possible values of \(B\)

Since \(\sin(B)\approx0.666\), \(B=\sin^{- 1}(0.666)\approx42^{\circ}\) (using the inverse - sine function for the acute angle). The other possible value of \(B\) is \(B = 180^{\circ}-42^{\circ}=138^{\circ}\) (because \(\sin(\theta)=\sin(180^{\circ}-\theta)\) for \(0^{\circ}<\theta < 180^{\circ}\)).

Answer:

The possible measures of angle \(B\) from least to greatest are \(42^{\circ}\) and \(138^{\circ}\).