Answer:
B = 34.2°
C = 105.8°
c = 12.0
Step-by-step explanation:
Measure of Angle B
To find the measure of angle B in triangle ABC, we can use the Law of Sines:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Sines}} \\\\\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\\\\\textsf{where:}\\\phantom{ww}\bullet \;\textsf{$A, B$ and $C$ are the angles.}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides opposite the angles.}\end{array}}[/tex]
In this case:
- A = 40°
- a = BC = 8
- b = AC = 7
Substitute these values into the relevant parts of the equation:
[tex]\dfrac{\sin 40^{\circ}}{8}=\dfrac{\sin B}{7}[/tex]
Now, solve for B:
[tex]\sf \dfrac{7\sin 40^{\circ}}{8}=\sin B \\\\\\ B=\sin^{-1}\left(\dfrac{7\sin 40^{\circ}}{8}\right) \\\\\\ B=34.2246502093...^{\circ} \\\\\\B=34.2^{\circ}\; (nearest\;tenth)[/tex]
Therefore, the measure of angle B rounded to the nearest tenth is:
[tex]\LARGE\text{$\sf B=\boxed{\sf 34.2}^{\circ}$}[/tex]
[tex]\dotfill[/tex]
Measure of Angle C
As the interior angles of a triangle always sum to 180°, then:
[tex]\sf A+B+C=180^{\circ}[/tex]
Substitute the measure of angle A and angle B into the equation and solve for C:
[tex]\sf 40^{\circ}+34.2246502093...^{\circ} +C=180^{\circ} \\\\74.2246502093...^{\circ} +C=180^{\circ} \\\\C=180^{\circ}-74.2246502093...^{\circ} \\\\C=105.775349790...^{\circ} \\\\C=105.8^{\circ}\; (nearest\;tenth)[/tex]
Therefore, the measure of angle C rounded to the nearest tenth is:
[tex]\LARGE\text{$\sf C=\boxed{\sf 105.8}^{\circ}$}[/tex]
[tex]\dotfill[/tex]
Length of Side c
To find the length of side c, we can use the Law of Sines again, ensuring we use the exact value of angle C:
[tex]\sf \dfrac{\sin 40^{\circ}}{8}=\dfrac{\sin 105.775349790...^{\circ}}{c} \\\\\\ c=\dfrac{8\sin 105.775349790...^{\circ}}{\sin 40^{\circ}} \\\\\\ c=11.977020494757... \\\\\\c=12.0\; (nearest\;tenth)[/tex]
Therefore, the length of side c rounded to the nearest tenth is:
[tex]\LARGE\text{$\sf c=\boxed{\sf 12.0}$}[/tex]
