Question

law of sines: \\(\frac{sin(a)}{a} = \frac{sin(b)}{b} = \frac{sin(c)}{c}\\) in \\(\triangle bcd\\), \\(d = 3\\), \\(b = 5\\), and \\(m angle d = 25^circ\\). what are the possible approximate measures of angle \\(b\\)? \\(\bigcirc\\) only \\(90^circ\\) \\(\bigcirc\\) only \\(155^circ\\) \\(\bigcirc\\) \\(20^circ\\) and \\(110^circ\\) \\(\bigcirc\\) \\(45^circ\\) and \\(135^circ\\)

Explanation:

Step1: Apply Law of Sines

$\frac{\sin(B)}{b} = \frac{\sin(D)}{d}$

Step2: Substitute given values

$\frac{\sin(B)}{5} = \frac{\sin(25^\circ)}{3}$

Step3: Solve for $\sin(B)$

$\sin(B) = \frac{5 \times \sin(25^\circ)}{3}$
$\sin(B) \approx \frac{5 \times 0.4226}{3} \approx 0.7043$

Step4: Find principal angle

$m\angle B \approx \arcsin(0.7043) \approx 45^\circ$

Step5: Find supplementary angle

$180^\circ - 45^\circ = 135^\circ$

Step6: Verify validity

Check $45^\circ + 25^\circ = 70^\circ < 180^\circ$, $135^\circ + 25^\circ = 160^\circ < 180^\circ$, both are valid.

Answer:

D. 45° and 135°