Question

find the missing parts of the triangle. round angles to one decimal place and sides to the nearest centimeter.
a = 111.9°
a = 1,267 cm
b = 1,367 cm
a. b = 31.4°, c = 36.7°, c = 1,279 cm
b. b = 32.8°, c = 35.3°, c = 1,279 cm
c. no such triangle exists.
d. b = 36.7°, c = 31.4°, c = 1,279 cm

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Substituting the given values \(a = 1267\), \(A=111.9^{\circ}\), and \(b = 1367\) into the formula, we get \(\frac{1267}{\sin111.9^{\circ}}=\frac{1367}{\sin B}\).
Cross - multiplying gives \(1267\sin B=1367\sin111.9^{\circ}\).
Then \(\sin B=\frac{1367\sin111.9^{\circ}}{1267}\).
Calculating \(\sin111.9^{\circ}\approx0.93\), we have \(\sin B=\frac{1367\times0.93}{1267}\approx1.01\).

Step2: Analyze the value of \(\sin B\)

Since the range of the sine function is \([- 1,1]\), and \(\sin B\approx1.01>1\), no such angle \(B\) exists.

Answer:

C. No such triangle exists.