Maximizing and Minimizing the Expression cos x - sin x: A Comprehensive Guide

Understanding how to find the maximum and minimum values of trigonometric expressions like cos x - sin x is essential for advancing your knowledge in calculus and higher mathematics. This article will walk you through the process step-by-step, detailing the mathematical reasoning and providing clear examples.

Introduction to Trigonometric Functions

Before we delve into the main topic, it's important to briefly review the properties of the cosine and sine functions. Both cos x and sin x are periodic functions with a period of (2pi). They oscillate between -1 and 1, and their derivatives, which are useful in finding critical points, are given by: ( frac{d}{dx} cos x -sin x ) ( frac{d}{dx} sin x cos x )

Deriving the Critical Points

To find the maximum and minimum values of the expression cos x - sin x, we need to start by finding its derivative and setting it to zero to find the critical points. The derivative of cos x - sin x is:

[ frac{d}{dx} (cos x - sin x) -sin x - cos x ]

Setting this derivative equal to zero:

[ -sin x - cos x 0 ]

Consequently, we get:

[ sin x -cos x ]

To solve this equation, we can divide both sides by cos x (assuming cos x ≠ 0):

[ tan x -1 ]

From trigonometry, we know that ( tan x -1 ) at:

[ x 135^circ text{ or } x 315^circ text{ (in radians, } x frac{3pi}{4} text{ or } x frac{7pi}{4} text{) } ]

Evaluating the Function at Critical Points

Now that we have the critical points, we evaluate the function cos x - sin x at these points to find the maximum and minimum values.

For ( x 135^circ ):

[ cos 135^circ -frac{sqrt{2}}{2}, quad sin 135^circ frac{sqrt{2}}{2} ]

[ cos 135^circ - sin 135^circ -frac{sqrt{2}}{2} - frac{sqrt{2}}{2} -sqrt{2} ]

For ( x 315^circ ):

[ cos 315^circ frac{sqrt{2}}{2}, quad sin 315^circ -frac{sqrt{2}}{2} ]

[ cos 315^circ - sin 315^circ frac{sqrt{2}}{2} - left( -frac{sqrt{2}}{2} right) sqrt{2} ]

Conclusion: Maximum and Minimum Values

Therefore, the maximum value of cos x - sin x is ( sqrt{2} ) and the minimum value is ( -sqrt{2} ). These values represent the extremes of the function over its domain.

Frequently Asked Questions (FAQ)

Q: Is there any other method to find the maximum and minimum values of cos x - sin x?

A: While the method we've used is straightforward, you can also use the fact that cos x - sin x can be rewritten as ( sqrt{2} cos left( x frac{pi}{4} right) ), which simplifies the problem and helps in visualizing the function more clearly.

Q: What are some practical applications of finding the maximum and minimum values of trigonometric expressions?

A: These concepts are used in various fields such as physics, engineering, and signal processing, where periodic functions play a crucial role in modeling oscillatory behavior and waveforms.

Q: How can one practice solving similar problems?

A: Practice is key. Try solving a variety of problems involving trigonometric expressions, and always check your solutions with graphing tools or calculators to ensure accuracy.