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Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

Short Answer

Expert verified

Ans: Quadrants came into account as the resultant absolute values are significant since tanuis positive for uin the first quadrant and negative for uin the second quadrant. That's why in this case unit circle is considered.

Step by step solution

01

Step 1. Given information:

x=asinux=atanu

02

Step 2. Solving the trigonometric substitution:

In the trigonometric substitution of x=asinu and x=atanu, 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=asecu, quadrants came into account as the resultant absolute values are significant since tanu 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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