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Q. 16

Expert-verifiedFound in: Page 464

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution with$x=a\mathrm{sin}u$ or $x=a\mathrm{tan}u$? Why do we need to think about the unit circle after trigonometric substitution with $x=asecu$?

Ans: Quadrants came into account as the resultant absolute values are significant since $\mathrm{tan}u$ is positive for $u$ in the first quadrant and negative for $u$ in the second quadrant. That's why in this case unit circle is considered.

$x=a\mathrm{sin}u\phantom{\rule{0ex}{0ex}}x=a\mathrm{tan}u$

In the trigonometric substitution of $x=a\mathrm{sin}u$ and $x=a\mathrm{tan}u$, the choice of quadrant will never be an issue. That's why the unit circle is never considered. But in the trigonometric substitution of $x=a\mathrm{sec}u$, quadrants came into account as the resultant absolute values are significant since $\mathrm{tan}u$ is positive for $u$ in the first quadrant and negative for $u$ in the second quadrant. That's why in this case unit circle is considered.

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