Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Find angle Q

The sum of angles in a triangle is $180^\circ$.
$\angle Q = 180^\circ - 45^\circ - 82^\circ = 53^\circ$

Step2: Find side $s$ via Law of Sines

Relate $s$, $\angle S$, $r$, $\angle R$.
$\frac{s}{\sin(45^\circ)} = \frac{7}{\sin(82^\circ)}$
$s = \frac{7\sin(45^\circ)}{\sin(82^\circ)} \approx \frac{7 \times 0.7071}{0.9903} \approx 5.0$

Step3: Find side $q$ via Law of Sines

Relate $q$, $\angle Q$, $r$, $\angle R$.
$\frac{q}{\sin(53^\circ)} = \frac{7}{\sin(82^\circ)}$
$q = \frac{7\sin(53^\circ)}{\sin(82^\circ)} \approx \frac{7 \times 0.7986}{0.9903} \approx 5.6$

Answer:

$\angle Q = 53^\circ$, $s \approx 5.0$, $q \approx 5.6$