Solution to Trig Inequality in [0,-1]

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SUMMARY

The discussion centers on proving that the equation x = sin(πx) + cos(πx) has a solution in the interval [0, -1]. Participants clarify that the function can be reformulated as f(x) = x - sin(πx) - cos(πx) = 0. Evaluating f(0) yields -1 and f(1) yields 2, indicating a change in sign, which confirms the existence of a solution in the specified interval according to the Intermediate Value Theorem.

PREREQUISITES
  • Understanding of trigonometric functions, specifically sine and cosine.
  • Familiarity with the Intermediate Value Theorem in calculus.
  • Basic knowledge of function evaluation and sign changes.
  • Ability to manipulate and analyze inequalities.
NEXT STEPS
  • Study the Intermediate Value Theorem in more depth.
  • Explore the properties of trigonometric functions, particularly their ranges and behaviors.
  • Learn about function transformations and their implications on solutions.
  • Investigate numerical methods for finding roots of equations, such as the bisection method.
USEFUL FOR

Students studying calculus, particularly those tackling trigonometric equations and inequalities, as well as educators looking for examples of the Intermediate Value Theorem in action.

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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?
 
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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?
 
skateza said:

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.
 

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