Given displaystyle csc(t)=[-12/7] and [(-pi/2)<t<(pi/2)]. Find sint, cost and tant. Give exact answers without decimals.

Given $$\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}\ \text{and}\ \displaystyle{\left[{\left(-\frac{\pi}{{2}}\right)}<{t}<{\left(\frac{\pi}{{2}}\right)}\right]}$$ .
Find $$\sin\ t,\ \cos\ t\ \text{and}\ \tan\ t.$$ Give exact answers without decimals.

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Malena

To determine the value of $$\sin\ t,\ \cos\ t\ \text{and}\ \tan\ t.$$
Given:
$$\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}{\quad\text{and}\quad}{\left[{\left(-\frac{\pi}{{2}}\right)}<{t}<{\left(\frac{\pi}{{2}}\right)}\right]}$$
$$\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}$$
Using trigonometric identity, $$\displaystyle \sin{{t}}=\frac{1}{ \csc{{t}}}$$
$$\displaystyle \sin{{t}}=\frac{1}{{\frac{{-{12}}}{{{7}}}}}$$
$$\displaystyle \sin{{t}}=-\frac{7}{{12}}$$
$$\displaystyle{{\cos}^{2}{t}}={1}-{{\sin}^{2}{t}}$$
$$\displaystyle \cos{{t}}=\pm\sqrt{{{1}-{\left(-\frac{7}{{12}}\right)}^{2}}}$$
$$\displaystyle \cos{{t}}=\pm\sqrt{{{1}-\frac{49}{{144}}}}$$
$$\displaystyle \cos{{t}}=\pm\sqrt{{\frac{95}{{144}}}}$$
Since, $$\displaystyle-\frac{\pi}{{2}}\le{t}\le\frac{\pi}{{2}}$$
$$\displaystyle\Rightarrow \cos{{t}}=\frac{\sqrt{{95}}}{{12}}$$
$$\displaystyle \tan{{t}}=\frac{{ \sin{{t}}}}{{ \cos{{t}}}}$$
$$\displaystyle \tan{{t}}=\frac{{{\left(-\frac{7}{{12}}\right)}}}{{{\left(\frac{\sqrt{{95}}}{{12}}\right)}}}$$
$$\displaystyle \tan{{t}}=-\frac{7}{\sqrt{{95}}}$$
Thus,
$$\displaystyle \sin{{t}}=-\frac{7}{{12}},$$
$$\displaystyle \cos{{t}}=\frac{\sqrt{{95}}}{{12}},$$
$$\displaystyle \tan{{t}}=-\frac{7}{\sqrt{{95}}}$$