Write the first trigonometric function in terms of the second for \theta i

banganX 2021-10-28 Answered
Write the first trigonometric function in terms of the second for \(\displaystyle\theta\) in the given quadrant
\(\displaystyle{\tan{\theta}},\ {\cos{\theta}};\ \theta\ \text{ in Quadrant II}\)

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Expert Answer

i1ziZ
Answered 2021-10-29 Author has 4496 answers
Since \(\displaystyle{\tan{\theta}}={\frac{{{\sin{\theta}}}}{{{\cos{\theta}}}}}\), we need to write \(\displaystyle{\sin{\theta}}\) in terms of \(\displaystyle{\cos{\theta}}\).
\(\displaystyle{\sin{\theta}}=\pm\sqrt{{{1}-{{\cos}^{{2}}\theta}}}\)
and since \(\displaystyle{\sin{\theta}}\) is negative in Quadrant III, the negative sign applies here. Thus
\(\displaystyle{\tan{\theta}}={\frac{{{\sin{\theta}}}}{{{\cos{\theta}}}}}={\frac{{-\sqrt{{{1}-{{\cos}^{{2}}\theta}}}}}{{{\cos{\theta}}}}}\)
Result: \(\displaystyle{\frac{{-\sqrt{{{1}-{{\cos}^{{2}}\theta}}}}}{{{\cos{\theta}}}}}\)
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