Answer:
[tex]\bold{cot(t) =\dfrac{12}{5}}[/tex]
Step-by-step explanation:
Given that:
[tex]Cosec (t) = -\frac{13}5[/tex]
for [tex]\pi < t < \frac{3 \pi}2[/tex]
That means, angle [tex]t[/tex] is in the 3rd quadrant.
To find:
Value of cot(t)
Solution:
First of all, let us recall what trigonometric ratios are positive and what trigonometric ratios are negative in 3rd quadrant.
In 3rd quadrant, tangent and cotangent are positive.
All other trigonometric ratios are negative.
Let us have a look at the following identity:
[tex]cosec^2\theta -cot^2\theta =1[/tex]
here, [tex]\theta =t[/tex]
So, [tex]cosec^2t-cot^2t=1[/tex]
[tex]\Rightarrow (-\dfrac{13}{5})^2-cot^2t=1\\\Rightarrow (\dfrac{169}{25})-cot^2t=1\\\Rightarrow \dfrac{169}{25}-1=cot^2t\\\Rightarrow \dfrac{169-25}{25}=cot^2t\\\Rightarrow \dfrac{144}{25}=cot^2t\\\Rightarrow cot(t)=\pm\sqrt{\dfrac{144}{25}}\\\Rightarrow cot(t)=\pm\dfrac{12}{5}[/tex]
But, angle [tex]t[/tex] is in 3rd quadrant, so value of
[tex]\bold{cot(t) =\dfrac{12}{5}}[/tex]
