Solution to Trig Inequality in [0,-1]

[skateza]

Homework Statement

Show that x=sin(piex) + cos(piex) has a solution in [0,-1]

The Attempt at a Solution

well i know that 0 <= x <= 1
therefore 0 <= sin(piex) + cos(piex) <= 1

But how do i go about solving this inequality?

 

[Dick]

You don't go about solving it. You just prove it has a solution. Change the equation into f(x)=x-sin(pi*x)-cos(pi*x)=0. f(0)=(-1). f(1)=2. Does that suggest anything?

 

[HallsofIvy]

Homework Statement

Show that x=sin(piex) + cos(piex) has a solution in [0,-1]

The Attempt at a Solution

well i know that 0 <= x <= 1
therefore 0 <= sin(piex) + cos(piex) <= 1

But how do i go about solving this inequality?

That certainly is NOT true! What is sin(pi)+ cos(pi)? Look at the signs of sin(pix)+ cos(pix)- x at 0 and 1.