Calc. Practice Problems #6 of 16?

Question

Hello. I have just uploded onto Photobucket 16 calculus practice problems, all of which will be of similar content to my eventual final exam. Now, an answer to these questions would be nice, but is not necessary. Instead, I am concerned with and interested in THE METHOD as to how one could arrive to the correct answer, as well as an EXPLANATION as to why each step needs to be performed in relation to the ones that follow. Anybody who can break down and explain the contents of these problems to me in a SIMPLISTIC way would just be amazing in my book. So thank you very much should you decide to help enlighten me, as I appreciate this more then the words in this post could even begin to convey!

Question #6: http://www.i138.photobucket.com/albums/q271/Link3324/math6.jpg

By the way, This is NOT for any kind of required assignment. It is a guide to work from and to understand before the actual test. Any help would be greatly appreciated. THANK YOU!

Answer 1

All one must do is sub the root of pi for x & for y, this means that you end up with an e to the power of 0 (which is 1)...

Now Sin (2 * root pi * root pi) is Sin (2 pi), & the Cos likewise ends up Cos (2 pi)... & the stuff on the right side of the two equals signs is or are right...

So you've shown how the two equations are satisfied by making x equal root of pi & y equal root of pi... if the two equations are satisfied by the same (x, y) pairing or coordinate, then the two curves must 'intersect' @ that point...

Answer 2

If you plug in the coordinates (√π,√π) you will see that it is indeed an intersection of the curves. To show that the curves are orthogonal, take the derivative of each curve to get the slope at (√π,√π). If the slopes are negative reciprocals of each other, the curves are orthogonal.