When can you use a triangle to represent trig substitution?

Question

I apologize if this seems like a newbie question but it's been really messing with my head. I'm reading about using trig substitution in integration and everyone seems to simply construct a right triangle to get what the trig functions will be (in terms of the original variable). Shouldn't that only work for cases where the angle is acute and positive? What about the cases where the angle is obtuse or where some trig functions might be negative. As far as I know, using the right triangle will always give a positive cos/sin/tan (because sides are always positive). Can you please explain in as much detail as possible how to correctly use a right triangle in cases where the angle is obtuse or one of the trig functions is negative. Thanks a lot in advance.

Answer 1

Drop an altitude to one of the sides.

Answer 2

This has nothing to do with triangles. A typical example is as follows:

Answer 3

When you use a trigonometric substitution in integration, you are using the trig function as a mathematical function and the substitution remains true whatever the value the variables take. You are not using the trig function as you do in solving a triangle.

For example, when you need to integrate a form which has sqrt(a² - x²), you might make the substitution

x=a sin u and wind up with cos u in the form. Then when you have evaluated the integral in terms of u, you have to convert back with careful use of the arcsin function.

What is at the heart of your question is how do we make the connection between the definitions of the trig functions we learned in elementary trig with the trig function as used in mathematical analysis. That, my dear questioner, is a much deeper question and depends on how you define the trig functions. If you are ready for it, get the textbook "Calculus" by M. Spivak and read with understanding the chapter on the trig functions. When you have done that you will have the whole answer to your question!