please, help me with this problem...?

Question

solve the equation:
sin(cosx)=cos(sinx)

Answer 1

In the 1st quadrant, the sine function is strictly increasing and the cosine is strictly decreasing. So, in this quadrant, both the functions sin(cos) and cos(sin) are strictly decreasing.

At x = 0 we have sin(cos(0)) = sin(1)=~ 0,841 < 1 = cos(sin(0)) (in radians). We can get to this conclusion, as well, observing that 1 < pi/2 and, therefore, sin(1) < sin(pi/2) =1.

In order to have sin(cos(x)) = cos(sin(x)), we must have sin(cos(x)) = sin(pi/2) - sin(x)), because the sines of 2 arcs are equal if, and only if, their determinations are the same or their sum is pi.

So, in the 1st case we must have cos(x) = 2*k*pi + pi/2 - sin(x) for some integer k. It follows that sen(x) + cos (x) = sqrt(2)* sin(pi/4 + x) = (2k+1/2) * pi. But, for every x, |sqrt(2) sin(pi/4 + x)| <= sqrt(2) and, for every integer k, |(2k+1/2) * pi| >= pi/2 > sqrt(2). Hence, the 1st case never happens.

In the second case we must have cos (x) + pi/2 - sin(x)= 2*k*pi + pi for some integer k. So, cos(x) - sin(x) = sqrt(2)* cos(pi/4 +x) = (2k+1/2)*pi. But for every real x and every integer k, |sqrt(2)* cos(pi/4 +x)| <= sqrt(2) and |(2k+1/2) * pi| >= pi/2 > sqrt(2), so that the 2nd case never happens, either.

So, it follows that sin(cos(x)) <> cos(sin(x)) for every real x. And you equation has no real roots. (but it can have complex roots)

In addition, from the continuity of the functions sin and cos and, so, of the composites sin(cos) and cos(sin), it follows that, since sin(cos(x)) < cos(sin(x)) for x = 0, then sin(cos(x)) < cos(sin(x)) for every real x. If there were some inversion in the inequality, the 2 curves, in virtue of their continuities, would intersect at some point, which, as we have seen, never happens.

You can confirm such conclusion by means of a chart.

Answer 2

The only angles with their sin and cos equal are 45 (pi/4) and 215 (5pi/4)... therefore, if we use 45 degrees...

sin(45) = cos(45), and sin(cosx)=cos(sinx)...

cosx = 45, and sinx = 45...

but this is not possible since the range of the sin and cos graph is -1>= y >= 1, where y is the value (in this case, 45)...

Answer 3

sin= you going to have to pay attention in class
cos= better hit the book

Answer 4

the question is incorrect !! check with calcalation!
try
Sin ( cos (30) ) = 0.0151
Cos ( Sin (30) ) = 0.9999

Answer 5

I have done this exercise in word and upload to mediafire.com!This link:----->http://www.mediafire.com/?ajdq9omjumm

Answer 6

Gotta calulater? THAN DO IT YOURSELF!! lol