Step-by-step explanation:
In trigonometry, the law of sines is an equation that relates the length of the sides of a triangle (any type of triangle) to the sines of its angles. This is expressed mathematically as
[tex]\displaystyle\frac{a}{\sin \alpha} \ = \ \displaystyle\frac{b}{\sin \beta} \ = \ \displaystyle\frac{c}{\sin \gamma}[/tex],
where [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are the lengths of the sides of the triangle and [tex]\alpha[/tex], [tex]\beta[/tex], and [tex]\gamma[/tex] are the opposite angles (as shown in the figure below). Likewise, the law is sometimes written as it reciprocals,
[tex]\displaystyle\frac{\sin \alpha}{a} \ = \ \displaystyle\frac{\sin \beta}{b} \ = \ \displaystyle\frac{\sin \gamma}{c}[/tex].
Therefore,
[tex]\displaystyle\frac{\sin A}{BC} \ = \ \displaystyle\frac{\sin B}{AC} \\ \\ \\ \sin A \ = \ \displaystyle\frac{\sin B \ \times \ BC}{AC} \\ \\ \\\sin A \ = \ \displaystyle\frac{\sin \left(90^{\circ}\right) \ \times \ \sqrt{\left(10\right)^{2} \ - \ \left(\sqrt{78}\right)^{2}}}{10} \\ \\ \\ \sin A \ = \ \dsiplaystyle\frac{\sqrt{22}}{10} \\ \\ \\ \sin A \ = \ 0.47 \ \ \ \ (\text{nearest hundredth})[/tex]
Alternatively, you can solve this question using the definition of the trigonometric function sine. For the angle [tex]\theta[/tex] in the figure below,
[tex]\sin \theta \ = \ \displaystyle\frac{\text{opposite}}{\text{hypothenuse}}[/tex].
Therefore,
[tex]\sin A \ = \ \displaystyle\frac{BC}{AC} \\ \\ \\ \sin A \ = \ \displaystyle\frac{\sqrt{\left(10\right)^{2} \ - \ \left(\sqrt{78}\right)^{2}}}{10} \\ \\ \\ \sin A \ = \ \displaystyle\frac{\sqrt{22}}{10} \\ \\ \\ \sin A \ = \ 0.47 \ \ \ \ (\text{nearest hundredth})[/tex]
