 # Given displaystyle csc(t)=[-12/7] and [(-pi/2)<t<(pi/2)]. Find sint, cost and tant Marvin Mccormick 2021-02-06 Answered

Given .
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To determine the value of
Given:
$\mathrm{csc}\left(t\right)=\left[\frac{-12}{7}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left[\left(-\frac{\pi }{2}\right)
$\mathrm{csc}\left(t\right)=\left[\frac{-12}{7}\right]$
Using trigonometric identity, $\mathrm{sin}t=\frac{1}{\mathrm{csc}t}$
$\mathrm{sin}t=\frac{1}{\frac{-12}{7}}$
$\mathrm{sin}t=-\frac{7}{12}$
${\mathrm{cos}}^{2}t=1-{\mathrm{sin}}^{2}t$
$\mathrm{cos}t=±\sqrt{1-{\left(-\frac{7}{12}\right)}^{2}}$
$\mathrm{cos}t=±\sqrt{1-\frac{49}{144}}$
$\mathrm{cos}t=±\sqrt{\frac{95}{144}}$
Since, $-\frac{\pi }{2}\le t\le \frac{\pi }{2}$
$⇒\mathrm{cos}t=\frac{\sqrt{95}}{12}$
$\mathrm{tan}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}$
$\mathrm{tan}t=\frac{\left(-\frac{7}{12}\right)}{\left(\frac{\sqrt{95}}{12}\right)}$
$\mathrm{tan}t=-\frac{7}{\sqrt{95}}$
Thus,
$\mathrm{sin}t=-\frac{7}{12},$
$\mathrm{cos}t=\frac{\sqrt{95}}{12},$
$\mathrm{tan}t=-\frac{7}{\sqrt{95}}$