Question

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

Short answer

Ans: Quadrants came into account as the resultant absolute values are significant since $\tan u$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.

Step-by-step solution

Step 1. Given information:

$$\begin{gathered} x=a\sin u \\ x=a\tan u \end{gathered}$$

Step 2. Solving the trigonometric substitution:

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