find sin(2x), cos(2x), and tan(2x) from the given information.\n\ncsc(x) = 8, tan(x) < 0\n\nsin(2x)…

Explanation:

Step1: Determine the quadrant of (x)

Since (\csc(x)=\frac{1}{\sin(x)} = 8>0) and (\tan(x)<0), (x) is in the second quadrant.

Step2: Find (\sin(x))

Given (\csc(x) = 8), then (\sin(x)=\frac{1}{8})

Step3: Find (\cos(x))

Using the identity (\sin^{2}(x)+\cos^{2}(x) = 1), we have (\cos^{2}(x)=1-\sin^{2}(x)=1 - (\frac{1}{8})^{2}=1-\frac{1}{64}=\frac{63}{64}). Since (x) is in the second quadrant, (\cos(x)=-\frac{3\sqrt{7}}{8})

Step4: Find (\sin(2x))

Using the double - angle formula (\sin(2x)=2\sin(x)\cos(x)) Substitute (\sin(x)=\frac{1}{8}) and (\cos(x)=-\frac{3\sqrt{7}}{8}) (\sin(2x)=2\times\frac{1}{8}\times(-\frac{3\sqrt{7}}{8})=-\frac{3\sqrt{7}}{32})

Step5: Find (\cos(2x))

Using the double - angle formula (\cos(2x)=1 - 2\sin^{2}(x)) Substitute (\sin(x)=\frac{1}{8}) (\cos(2x)=1-2\times(\frac{1}{8})^{2}=1-\frac{2}{64}=\frac{31}{32})

Step6: Find (\tan(2x))

Using the formula (\tan(2x)=\frac{\sin(2x)}{\cos(2x)}) Substitute (\sin(2x)=-\frac{3\sqrt{7}}{32}) and (\cos(2x)=\frac{31}{32}) (\tan(2x)=-\frac{3\sqrt{7}}{31})

Answer:

(\sin(2x)=-\frac{3\sqrt{7}}{32}) (\cos(2x)=\frac{31}{32}) (\tan(2x)=-\frac{3\sqrt{7}}{31})