Find the exact value of each expression. (tan)(17pi)/12

iohanetc 2021-09-03 Answered
Find the exact value of each expression.
\(\displaystyle{\left({\tan}\right)}\frac{{{17}\pi}}{{12}}\)

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Expert Answer

Clara Reese
Answered 2021-09-04 Author has 6307 answers
1. Split up the argument of the trigonometric function and then use the Sum formula for the Tangent function, \(\displaystyle{\tan{{\left({u}+{v}\right)}}}=\frac{{{\tan{{\left({u}\right)}}}+{\tan{{\left({v}\right)}}}}}{{{1}-{\tan{{\left({u}\right)}}}{\tan{{\left({v}\right)}}}}}\) , to find the value of given expression.
2. \(\displaystyle{\tan{{\left(\frac{{{17}\pi}}{{12}}\right)}}}={\tan{{\left(\frac{{{9}\pi}}{{12}}+\frac{{{8}\pi}}{{12}}\right)}}}\)
\(\displaystyle={\tan{{\left(\frac{{{3}\pi}}{{4}}+\frac{{{2}\pi}}{{3}}\right)}}}\)
\(\displaystyle=\frac{{{\tan{{\left(\frac{{{3}\pi}}{{4}}\right)}}}+{\tan{{\left(\frac{{{2}\pi}}{{3}}\right)}}}}}{{{1}-{\left({\tan{{\left(\frac{{{3}\pi}}{{4}}\right)}}}{\tan{{\left(\frac{{{2}\pi}}{{3}}\right)}}}\right.}}}\)
\(\displaystyle=\frac{{-{1}+{\left(-\sqrt{{3}}\right)}}}{{{1}-{\left(-{1}\right)}\cdot{\left(-\sqrt{{3}}\right)}}}\)
\(\displaystyle=\frac{{-{1}-\sqrt{{3}}}}{{{1}-\sqrt{{3}}}}\)
\(\displaystyle=\frac{{{\left(-{1}-\sqrt{{3}}\right)}{\left({1}+\sqrt{{3}}\right)}}}{{{\left({1}-\sqrt{{3}}\right)}{\left({1}+\sqrt{{3}}\right)}}}\)
\(\displaystyle=\frac{{-{1}-{2}\sqrt{{3}}-{3}}}{{{1}-{3}}}\)
\(\displaystyle=\frac{{{4}-{2}\sqrt{{3}}}}{{-{{2}}}}\)
\(\displaystyle={2}+\sqrt{{3}}\)
Result : \(\displaystyle{\tan{{\left(\frac{{{17}\pi}}{{12}}\right)}}}={2}+\sqrt{{3}}\)
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