ANSWER
[tex] \cot( \theta) = - \frac{4}{3} [/tex]
EXPLANATION
We were given that
[tex] \sin( \theta) = - \frac{3}{5} [/tex]
We need to find
[tex] \cot( \theta) .[/tex]
We know that,
[tex] \cot( \theta) = \frac{ \cos( \theta) }{ \sin( \theta) } [/tex]
We now need to find
[tex] \cos( \theta) [/tex]
using the Pythagorean identity or the right angle triangle.
According to the Pythagorean identity,
[tex] \cos^{2} \theta + \sin^{2} \theta = 1[/tex]
[tex] \cos^{2} \theta + {( \frac{ - 3}{5} )}^{2} = 1[/tex]
[tex] \cos^{2} \theta + \frac{9}{25} = 1[/tex]
[tex] \cos^{2} \theta = 1 - \frac{9}{25} [/tex]
[tex] \cos^{2} \theta = \frac{16}{25} [/tex]
[tex] \cos\theta = \pm \: \sqrt{ \frac{16}{25} } [/tex]
[tex] \cos\theta = \pm \: \frac{4}{5} [/tex]
Since we are dealing with the fourth quadrant,
[tex] \cos\theta = \frac{4}{5} [/tex]
This implies that,
[tex] \cot( \theta) = \frac{ \frac{4}{5} }{ - \frac{3}{5} } [/tex]
[tex] \cot( \theta) = - \frac{4}{3} [/tex]
The correct answer is D.
