Question

use the law of sines to find the length of side c, to the nearest millimeter, if \\(\angle a = 72\\) degrees and side \\(a = 1,246\\) mm. (1 point) \\(\bigcirc\\) 366 mm \\(\bigcirc\\) 3,835 mm \\(\bigcirc\\) 405 mm \\(\bigcirc\\) 385 mm

Explanation:

Step1: Identify triangle angles

$\angle B = 90^\circ$, $\angle A = 72^\circ$, so $\angle C = 180^\circ - 90^\circ - 72^\circ = 18^\circ$

Step2: Apply Law of Sines

Set up ratio: $\frac{c}{\sin\angle C} = \frac{a}{\sin\angle A}$

Step3: Solve for side $c$

Rearrange formula: $c = \frac{a \cdot \sin\angle C}{\sin\angle A}$
Substitute values: $c = \frac{1246 \cdot \sin(18^\circ)}{\sin(72^\circ)}$
Calculate: $\sin(18^\circ)\approx0.3090$, $\sin(72^\circ)\approx0.9511$
$c \approx \frac{1246 \times 0.3090}{0.9511} \approx \frac{385.014}{0.9511} \approx 405$

Answer:

405 mm