Answer:
The missing parts of the triangle are [tex]c \approx 11.2\,km[/tex], [tex]A\approx 21.3^{\circ}[/tex] and [tex]B \approx 38.8^{\circ}[/tex], respectively.
Step-by-step explanation:
A triangle is formed by three sides and three angles, a side and two angles are missing ([tex]c[/tex], [tex]A[/tex], [tex]B[/tex]).
The length of the missing side is found by the Law of Cosine:
[tex]c = \sqrt{a^{2}+b^{2}-2\cdot a \cdot b\cdot \cos C}[/tex] (1)
([tex]a = 4.7\,km[/tex], [tex]b = 8.1\,km[/tex], [tex]C = 119.90^{\circ}[/tex])
[tex]c = \sqrt{(4.7\,km)^{2}+(8.1\,km)^{2}-2\cdot (4.7\,km)\cdot (8.1\,km)\cdot \cos 119.90^{\circ}}[/tex]
[tex]c \approx 11.2\,km[/tex]
And the missing angles can be determined by the Law of Sine:
[tex]\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}[/tex]
([tex]a = 4.7\,km[/tex], [tex]b = 8.1\,km[/tex], [tex]c \approx 11.210\,km[/tex], [tex]C = 119.9^{\circ}[/tex])
[tex]A \approx \sin^{-1}\left(\frac{a}{c} \times \sin C \right)[/tex]
[tex]A \approx \sin^{-1} \left(\frac{4.7\,km}{11.210\,km}\times \sin 119.9^{\circ} \right)[/tex]
[tex]A\approx 21.3^{\circ}[/tex]
[tex]B \approx \sin^{-1} \left(\frac{b}{c}\times \sin C \right)[/tex]
[tex]B\approx \sin^{-1}\left(\frac{8.1\,km}{11.210\,km}\times \sin 119.9^{\circ} \right)[/tex]
[tex]B \approx 38.8^{\circ}[/tex]
The missing parts of the triangle are [tex]c \approx 11.2\,km[/tex], [tex]A\approx 21.3^{\circ}[/tex] and [tex]B \approx 38.8^{\circ}[/tex], respectively.
