The information we have is:
[tex]\sin x=-\frac{1}{3}[/tex]Step 1. Use the following trigonometric property between the sine and the cosine:
[tex]\sin ^2x+\cos ^2x=1[/tex]Solving for cos^2x by subtracting sin^2x to both sides:
[tex]\cos ^2x=1-\sin ^2x[/tex]Then, we take the square root of both sides:
[tex]\cos x=\pm\sqrt[]{1-\sin ^2x}[/tex]Step 2. Substitute the value of sinx to find cosx:
[tex]\cos x=\pm\sqrt[]{1-(-\frac{1}{3}^2)}[/tex]Solving the operations inside the square root:
[tex]\begin{gathered} \cos x=\pm\sqrt[]{1-\frac{1}{9}} \\ \\ \cos x=\pm\sqrt[]{\frac{8}{9}} \end{gathered}[/tex]We simplify this result considering the following:
[tex]\begin{gathered} \sqrt[]{\frac{8}{9}}=\frac{\sqrt[]{8}}{\sqrt[]{9}} \\ \text{And since} \\ \sqrt[]{8}=\sqrt[]{4\cdot2}=2\sqrt[]{2} \\ \sqrt[]{9}=3 \\ \end{gathered}[/tex]Using that, cosx is now as follows:
[tex]\cos x=\pm\frac{2\sqrt[]{2}}{3}[/tex]Step 3. Which sign are we going to use, the + or the - sign?
The solution to this can be found by considering the condition:
[tex]\tan x>0[/tex]the tangent of x has to be greater than 0 (it has to be a positive number).
The definition of tangent is as follows:
[tex]\tan x=\frac{\sin x}{\cos x}[/tex]Since sinx is already a negative number (-1/3) we conclude that, for tanx to be a positive number cox has to also be a negative number so the division of the two minus signs becomes a positive.
thus:
[tex]\cos x=-\frac{2\sqrt[]{2}}{3}[/tex]So that:
[tex]\tan x=\frac{\sin x}{\cos x}=\frac{-\frac{1}{3}}{\frac{-2\sqrt[]{2}}{3}}>0[/tex]Step 4. We already have the cosine of x, but we still need the cotangent of x.
Which is defined as follows:
[tex]\cot x=\frac{\cos x}{\sin x}[/tex]So we take our values for cosx and sinx and substitute them:
[tex]\cot x=\frac{\frac{-2\sqrt[]{2}}{3}}{\frac{-1}{3}}[/tex]To solve this division, we multiply the values in the extremes and the middle values as follows:
[tex]\cot x=\frac{\frac{-2\sqrt[]{2}}{3}}{\frac{-1}{3}}=\frac{-2\sqrt[]{2}\cdot3}{3\cdot(-1)}[/tex]Continue solving the operations:
[tex]\cot x=\frac{\frac{-2\sqrt[]{2}}{3}}{\frac{-1}{3}}=\frac{-6\sqrt[]{2}}{-3}[/tex]Finally, we divide -6 by -3, which gives us 2 as the result. And the final result for cotx is:
[tex]\cot x=2\sqrt[]{2}[/tex]Step 5. Diagram of the angle x.
To show where is angle x, we need to know the value of angle x.
For this, we use the given value for sinx:
[tex]\sin x=-\frac{1}{3}[/tex]And apply the inverse sine to find x:
[tex]\begin{gathered} x=\sin ^{-1}(-\frac{1}{3}) \\ x=-19.47 \end{gathered}[/tex]Since the angle is a negative number, in the coordinate plane it will be below the horizontal plane:
Answer:
[tex]\begin{gathered} \cot x=2\sqrt[]{2} \\ \cos x=-\frac{2\sqrt[]{2}}{3} \end{gathered}[/tex]