the equation cosx=x has a solution in what interval?...the answer is given as [0,pi/2] can someone please explain why?? thnks

That is not the correct answer. Any real number, positive or negative, has a cosine.

You do not even encounter the full range of cosine values when you limit the domain to [0,pi/2]

I suspect you may be using a substandard text or instructor

its not just cosx...its cosx=x

My mistake. I apologize for misreadng the problem.

cosx is confined to the range -1 to 1. Any solutions must first of all be within -1 < x < 1. In the x region -1 to 0, -57.3 to 0 degrees, cos x is positive so there can be no solution for negative x. There is also no solution for x > 1. The only solution that exists is around 0.738. They are correct in saying that the solution is in the domain [0,pi/2], but actually they could have better said it was between 0.73 and 0.74.

To state that [0,pi/2] is the only interval where a solution exists is incorrect and arbitrary. The solution clearly could not be beyond x = 1.

Your confusion about their stated answer is justified

To find the solution to the equation cos(x) = x and the interval in which it holds, we can follow these steps:

Step 1: Understand the equation. In this case, we have the equation cos(x) = x. We want to find the values of x that satisfy this equation.

Step 2: Graph the functions. Plot the graphs of the functions y = cos(x) and y = x on the same coordinate plane. This will help us visualize where they intersect.

Step 3: Observe the graphs. By looking at the graphs, we can see that the cosine function has a maximum value of 1 and the line y = x passes through the point (0, 0).

Step 4: Observe the interval limits. Notice that the interval given in the solution, [0, pi/2], includes the points where the two graphs intersect.

Step 5: Check the values in the interval. To determine if the equation holds for all the values in the interval [0, pi/2], we need to test some points within the interval.

Step 6: Test the interval limits. For the lower limit, x = 0, if we substitute it into the equation cos(x) = x, we get cos(0) = 0, which is true. This means that the equation holds for the lower limit.

For the upper limit, x = π/2, if we substitute it into the equation cos(x) = x, we get cos(π/2) = π/2, which is also true. Therefore, the equation holds for the upper limit as well.

Step 7: Conclusion. Since the equation cos(x) = x holds true for all the points in the interval [0, π/2], we can conclude that the solution to the equation cos(x) = x falls within that interval.

Therefore, the solution to the equation cos(x) = x is x ∈ [0, π/2].