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 is given by \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(A = 111.9^{\circ}\), \(a = 1267\) cm, and \(b = 1367\) cm. First, find \(\sin B\) using \(\frac{a}{\sin A}=\frac{b}{\sin B}\). So, \(\sin B=\frac{b\sin A}{a}\).
Substitute the values: \(\sin B=\frac{1367\times\sin(111.9^{\circ})}{1267}\).
Calculate \(\sin(111.9^{\circ})\approx0.928\). Then \(\sin B=\frac{1367\times0.928}{1267}\approx1.01\).
Since the range of the sine function is \([- 1,1]\), and \(\sin B\approx1.01>1\).

Answer:

C. No such triangle exists