Question

in the following triangle, $\theta = 60\degree$. find the values of the angles $b$ and $b$, which solve this ambiguous case.
a. $b = 70.27\degree$ or $b = 109.72\degree$
b. $b = 72.45\degree$ or $b = 107.55\degree$
c. $b = 69.87\degree$ or $b = 110.13\degree$
d. $b = 66.25\degree$ or $b = 113.75\degree$

Explanation:

Step1: Apply Law of Sines

$\frac{\sin B}{13} = \frac{\sin 60^\circ}{12.3}$

Step2: Solve for $\sin B$

$\sin B = \frac{13 \times \sin 60^\circ}{12.3}$
$\sin B = \frac{13 \times \frac{\sqrt{3}}{2}}{12.3} \approx 0.940$

Step3: Find acute angle $B$

$B = \arcsin(0.940) \approx 70.27^\circ$

Step4: Find obtuse angle $B'$

$B' = 180^\circ - 70.27^\circ = 109.73^\circ \approx 109.72^\circ$

Step5: Verify validity

Check $60^\circ + 109.72^\circ < 180^\circ$, which holds.

Answer:

A. $B = 70.27^\circ$ or $B' = 109.72^\circ$