Given the tangent functions of y = 1– 3 \tan (\frac{2x-\pi}{4}), find the

tricotasu 2021-09-29 Answered
Given the tangent functions of \(\displaystyle{y}={1}–{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\), find the Equation of all of its vertical asymptotes.

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Expert Answer

delilnaT
Answered 2021-09-30 Author has 7103 answers
Step 1
Given- \(\displaystyle{y}={1}–{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\)
To find- The equation of vertical asymptotes.
Concept Used- To find the vertical asymptotes of the function \(\displaystyle{y}={\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}}\), equate q(x) with 0 and solve accordingly.
Step 2
Explanation- Rewrite the given expression,
\(\displaystyle{y}={1}-{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\)
Simplify the above expression, we get,
\(\displaystyle{y}={1}-{\frac{{{3}}}{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}}}}\)
\(\displaystyle{y}={\frac{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}-{3}}}{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}}}}\)
For vertical asymptotes, equate denominator with zero, we get,
\(\displaystyle{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}={0}\)
\(\displaystyle{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}={0}\)
\(\displaystyle{2}{x}-\pi={0}\)
\(\displaystyle{2}{x}=\pi\)
\(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\)
So, the equation of the vertical asymptotes is \(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\).
Answer- Hence, the equation of the vertical asymptotes is \(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\).
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