Answer:
1. [tex] \theta = 61.9^\circ [/tex]
2. [tex] x = 23.7~cm [/tex]
Step-by-step explanation:
Problem 1.
Use the Law of Sines.
This is SSA.
[tex]\dfrac{a}{\sin A} = \dfrac{c}{\sin C}[/tex]
[tex] \dfrac{31.2~cm}{\sin 51^\circ} = \dfrac{35.4~cm}{\sin \theta} [/tex]
[tex] \sin \theta = \dfrac{35.4~cm \times \sin 51^\circ}{31.2~cm} [/tex]
[tex] \sin \theta = 0.88176 [/tex]
[tex] \theta = \sin^{-1} 0.88176 [/tex]
[tex] \theta = 61.9^\circ [/tex]
Problem 2.
m<A + m<B + m<C = 180°
m<A + 67° + 58° = 180°
m<A = 55°
Use the Law of Sines.
[tex]\dfrac{a}{\sin A} = \dfrac{c}{\sin C}[/tex]
[tex]\dfrac{x}{\sin 55^\circ} = \dfrac{24.5}{\sin 58^\circ}[/tex]
[tex]x = \dfrac{\sin 55^\circ \times 24.5}{\sin 58^\circ}[/tex]
[tex] x = 23.7~cm [/tex]
