Respuesta :
Out of the given choice, the equation represents [tex]\cot \theta=\frac{\sqrt{15}}{7}[/tex].
Answer: Option B
Step-by-step explanation:
We know, [tex]\csc \theta=\frac{1}{\sin \theta}[/tex]
[tex]\sin \theta=\frac{1}{\csc \theta}[/tex]
Given data:
[tex]\csc \theta=\frac{8}{7}[/tex]
So, now sin theta can express as
[tex]\sin \theta=\frac{7(\text { opposite })}{8(\text { Hypotenuse })}[/tex]
Sin theta defined by the ratio of opposite to the hypotenuse. In general, the adjacent can be calculated by,
[tex]\text {(opposite) }^{2}+(\text { adjacent })^{2}=(\text {Hypotenuse})^{2}[/tex]
[tex]7^{2}+(\text { adjacent })^{2}=8^{2}[/tex]
[tex](\text {adjacent})^{2}=8^{2}-7^{2}=64-49=15[/tex]
Taking square root, we get
[tex]\text { adjacent }=\sqrt{15}[/tex]
Also, we know the formula for cot theta,
[tex]\cot \theta=\frac{1}{\tan \theta}=\frac{1}{\left(\frac{\sin \theta}{\cos \theta}\right)}=\frac{\cos \theta}{\sin \theta}[/tex]
Cos theta denoted as the ratio of adjacent to the hypotenuse.
[tex]\cos \theta=\frac{\sqrt{15}(\text {Adjacent})}{8(\text {Hypotenuse})}[/tex]
Therefore, find now as below,
[tex]\cot \theta=\frac{\left(\frac{\sqrt{15}}{8}\right)}{\left(\frac{7}{8}\right)}=\frac{\sqrt{15}}{8} \times \frac{8}{7}=\frac{\sqrt{15}}{7}[/tex]
