Question

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

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asked 2021-02-06

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.

Expert Answers (1)

2021-02-07
To determine the value of \(\sin\ t,\ \cos\ t\ \text{and}\ \tan\ t.\)
Given:
\(\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}{\quad\text{and}\quad}{\left[{\left(-\frac{\pi}{{2}}\right)}<{t}<{\left(\frac{\pi}{{2}}\right)}\right]}\)</span>
\(\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}\)
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,
\(\displaystyle \sin{{t}}=-\frac{7}{{12}},\)
\(\displaystyle \cos{{t}}=\frac{\sqrt{{95}}}{{12}},\)
\(\displaystyle \tan{{t}}=-\frac{7}{\sqrt{{95}}}\)
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