Question

consider a triangle (abc) like the one below. suppose that (b = 44^circ), (c = 102^circ), and (c = 64). (the figure is not drawn to scale.)
solve the triangle.

round your answers to the nearest tenth.
if there is more than one solution, use the button labeled \or\.

(a = \box^circ), (a = \box), (b = \box)

Explanation:

Find the missing angle A

The sum of angles in any triangle is \(180^\circ\).
Given:

$$B = 44^\circ, \quad C = 102^\circ$$

We calculate \(A\):

$$A = 180^\circ - B - C$$

$$A = 180^\circ - 44^\circ - 102^\circ = 34^\circ$$

Apply Law of Sines for side a

Using the Law of Sines:

$$\frac{a}{\sin(A)} = \frac{c}{\sin(C)}$$

Substitute the known values \(A = 34^\circ\), \(C = 102^\circ\), and \(c = 64\):

$$a = \frac{c \cdot \sin(A)}{\sin(C)} = \frac{64 \cdot \sin(34^\circ)}{\sin(102^\circ)}$$

Using a calculator:

$$\sin(34^\circ) \approx 0.55919$$

$$\sin(102^\circ) \approx 0.97815$$

$$a \approx \frac{64 \cdot 0.55919}{0.97815} \approx 36.585 \approx 36.6$$

Apply Law of Sines for side b

Using the Law of Sines:

$$\frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

Substitute the known values \(B = 44^\circ\), \(C = 102^\circ\), and \(c = 64\):

$$b = \frac{c \cdot \sin(B)}{\sin(C)} = \frac{64 \cdot \sin(44^\circ)}{\sin(102^\circ)}$$

Using a calculator:

$$\sin(44^\circ) \approx 0.69466$$

$$b \approx \frac{64 \cdot 0.69466}{0.97815} \approx 45.451 \approx 45.5$$

Answer:

Consider a triangle \(ABC\) like the one below. Suppose that \(B=44^\circ\), \(C=102^\circ\), and \(c=64\). Solve the triangle.

Round your answers to the nearest tenth.

\(A =\) <blank>\(34\)</blank>\(^\circ\), \(a =\) <blank>\(36.6\)</blank>, \(b =\) <blank>\(45.5\)</blank>