Question

Explain how to know when to use the trigonometric substitutions $x=a\sin u,x=a\tan u,andx=a\sec u$, Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for $x$and $u$in each case?

Short answer

Ans: In the term of the Pythagorean identity, sine and cosine behave the same way. That's why we can always use $x=\sin u$instead of $x=\cos u$.

Step-by-step solution

Step 1. Given information.

given,

$x=a\sin u,x=a\tan u,\text{and}x=a\sec u$

Step 2. Solution:- 

In terms of the Pythagorean identity, sine and cosine behave the same way. That's why we can always use $x=\sin u$instead of $x=\cos u$.