Prove that |sinx-siny| ≤|x-y| and |cosx-cosy| ≤|x-y| for all x, y ∈ℝ.

Name: Problem 21, Chapter 7: A Course in Calculus and Real Analysis
Uploaded: 2021-09-16T16:08:56-08:00
Duration: 1 min 55 s
Channel: Eric Mockensturm
Description: Prove that |sinx-siny| ≤|x-y| and |cosx-cosy| ≤|x-y| for all x, y ∈ℝ.

step 1 So they want us to show that the absolute value of cosine of x minus cosine of y is less than or

Prove that |sinx-siny| ≤|x-y| and |cosx-cosy| ≤|x-y| for all...

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental result in calculus that states, for a function that is continuous on a closed interval and differentiable on the open interval, there exists a point in the open interval where the derivative equals the average rate of change over the interval. In the context of the inequalities involving sine and cosine, it provides the mechanism to relate the difference of the function values to its derivative, allowing us to bound the difference by the magnitude of the interval between the inputs.

Lipschitz Continuity

A function is Lipschitz continuous if there exists a constant L such that for all pairs of points, the absolute difference in function values is bounded by L times the absolute difference in the input values. Sine and cosine functions are examples of Lipschitz continuous functions with a Lipschitz constant of 1, meaning that the functions do not change faster than the rate of the change in the input, which is precisely what the provided inequalities convey.

Bounded Derivatives

The concept of bounded derivatives is important in ensuring that a function does not have abrupt changes in its slope. Specifically, the derivatives of sine and cosine are bounded by 1 in absolute value. This boundedness is crucial in applying the Mean Value Theorem because it guarantees that the average rate of change (as given by the derivative at some intermediate point) does not exceed 1, thus allowing us to deduce that the difference in the function values is at most the difference in the input values.

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