<p>Why are the following substitutions used:</p>
<p>x = a sin θ
x = a tan θ
x = a sec θ</p>
<p>as opposed to the following substitutions:</p>
<p>x = a cos θ
x = a cot θ
x = a csc θ</p>
<p>it really depends on the problems.
I think you should go back and review ;)</p>
<p>Wait...
What do you mean?</p>
<p>you can use the cos, cot and csc but all you are doing is adding extra negative signs and makes things more complicated than it has to be</p>
<p>Are you sure?
Can someone verify this?</p>
<p>Lets use sqrt(1-x^2) as and integrate it from (0,1)</p>
<p>if you let x=sinθ you should get the intergral<a href="0,%20pi/2">cos^2(θ)dθ</a> I let you check it over
now let x=cosθ and you should get -integral<a href="pi/2,0">sin^2(θ)dθ</a> or integral<a href="0,pi/2">sin^2(θ)dθ</a> </p>
<p>Both are equal to each other (check with a graphing calculator)</p>
<p>but I shouldn't say that cosθ and the rest are useless, if you are dealing with negative signs to to begin with it may end up being easier</p>
<p>There's some proof. I used to know it. It should be in your book. I'm sure you could easily find it online.</p>
<p>I suspect the answer lies partially with the derivative of inverse trig functions (a calc I topic).</p>
<p>d/dx[Arcsin x] = 1/sqrt(1 - x^2)
d/dx[Arccos x] = -1/sqrt(1 - x^2)</p>
<p>d/dx[Arctan x] = 1/(1 + x^2)
d/dx[Arccot x] = -1/(1 + x^2)</p>
<p>d/dx[Arcsec x] = 1/(abs(x)<em>sqrt(x^2 - 1))
d/dx[Arccsc x] = -1/(abs(x)</em>sqrt(x^2 - 1))</p>
<p>If you know these basic rules, trig substitution helps you to take antiderivatives of functions that look very much like these derivative results.</p>
<p>And so you use, x = a sin θ in the case where you have an (a^2 - x^2), x = a tan θ in the case where you have an (a^2 + x^2), and x = a sec θ in the case where you have an (x^2 - a^2).</p>
<p>In essence, though, viciouspoultry has the right idea. There's nothing mathematically wrong with using (a cos θ) en lieu of (a sin θ). I know that in the FDWK text for Calc I, they only show the derivative of Arcsin x, Arctan x, and Arcsec x, while allowing the student to show the related formulas for Arccos x, Arccot x, and Arccsc x. My guess is that's the reason why sin x, tan x, and sec x are used for trig substitution.</p>
<p>in summary, we use those because we just can't deal with negative signs.</p>