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Expert-verified Found in: Page 464 ### Calculus

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.

See the step by step solution

## Step 1. Given information:

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

## Step 2. Solving the trigonometric substitution:

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. ### Want to see more solutions like these? 