given the following information, find the value of \\( \\sin t \\) and \\( \\csc t \\). show all work.\n18)…

given the following information, find the value of \\( \\sin t \\) and \\( \\csc t \\). show all work.\n18) \\( \\cot t = \\frac { 7 } { 24 } \\) for \\( \\pi < t < \\frac { 3 \\pi } { 2 } \\)\n18. \\( \\sin t = \\)\n\\( \\csc t = \\)

given the following information, find the value of \\( \\sin t \\) and \\( \\csc t \\). show all work.\n18) \\( \\cot t = \\frac { 7 } { 24 } \\) for \\( \\pi < t < \\frac { 3 \\pi } { 2 } \\)\n18. \\( \\sin t = \\)\n\\( \\csc t = \\)

Answer

Explanation:

Step1: Use the Pythagorean identity (1+\cot^{2}t=\csc^{2}t)

Given (\cot t = \frac{7}{24}), then (1+\left(\frac{7}{24}\right)^{2}=\csc^{2}t). [ \begin{align*} 1+\frac{49}{576}&=\csc^{2}t\ \frac{576 + 49}{576}&=\csc^{2}t\ \frac{625}{576}&=\csc^{2}t \end{align*} ] Since (\pi<t<\frac{3\pi}{2}), (\csc t<0). So (\csc t=-\frac{25}{24})

Step2: Use the reciprocal identity (\sin t=\frac{1}{\csc t})

Since (\csc t =-\frac{25}{24}), then (\sin t=-\frac{24}{25})

Answer:

(\sin t=-\frac{24}{25}) (\csc t=-\frac{25}{24})