Step 1
Find the length of AC
In the right triangle ABC
Applying the Pythagoras Theorem
[tex]AC^{2}=AB^{2}+BC^{2}[/tex]
In this problem we have
[tex]AB=9\ units\\BC=16\ units[/tex]
Substitute
[tex]AC^{2}=9^{2}+16^{2}[/tex]
[tex]AC^{2}=337[/tex]
[tex]AC=\sqrt{337}\ units[/tex]
Step 2
Find the csc(A)
we know that
[tex]csc(A)=\frac{1}{sin(A)}[/tex]
[tex]sin(A)=\frac{BC}{AC}[/tex]
so
[tex]csc(A)=\frac{AC}{BC}[/tex]
substitute
[tex]csc(A)=\frac{\sqrt{337}}{16}[/tex]
therefore
the answer is
[tex]csc(A)=\frac{\sqrt{337}}{16}[/tex]
