Introduction

Understanding the minimum value of the expression cos x - sin x is crucial in calculus and trigonometry. This article explores this expression through three different methods, providing a comprehensive understanding of the concept and its applications.

Method 1: Calculus Approach

To find the minimum value of the expression cos x - sin x, we can use calculus. By differentiating the function and setting it to zero, we can identify critical points and determine the minimum value.

Step-by-Step Solution

Define the function: Take the derivative: Set the derivative equal to zero to find critical points: Solve for x and evaluate the function at these points to find the minimum value.

Let's proceed with these steps:

Given: f(x) cos x - sin x

Derivative: f'(x) -sin x - cos x

Setting the derivative equal to zero:

-sin x - cos x 0

Solving: This can be simplified by recognizing that -sin x - cos x -(sin x cos x). Setting sin x cos x 0 reveals that tan x -1. This equation has solutions like x 135° or -45°, or in radians, 3π/4 and -π/4.

Evaluating the function at these points:

x 135° (or 3π/4 in radians): cos(135°) - sin(135°) -1/√2 - 1/√2 -√2 x -45° (or -π/4 in radians): cos(-45°) - sin(-45°) 1/√2 1/√2 √2

The minimum value of cos x - sin x is therefore -√2.

Method 2: Trigonometric Identities

Another approach involves using trigonometric identities to simplify the expression. This method often leads to a more straightforward solution.

Step-by-Step Solution

Rewrite the expression using a trigonometric identity: Simplify to find the minimum value.

Rewrite:

cos x - sin x √2 cos (x - π/4)

Explanation: Here, cos (x - π/4) (1/√2)cos x - (1/√2)sin x.

The cosine function has a maximum value of 1 and a minimum value of -1. Therefore, the minimum value of cos (x - π/4) is -1.

Substituting back:

√2 cos(x - π/4) √2 * (-1) -√2

The minimum value of cos x - sin x is again -√2.

Method 3: Cauchy–Schwarz Inequality

A third approach uses the Cauchy–Schwarz inequality to find the minimum value. This approach leverages geometric intuition to simplify the problem.

Geometric Interpretation

Given: sin x - cos x cos x - sin x

Step 1: Represent the expression as a dot product of two vectors:

sin x - cos x sin x cos 1 - cos x sin 1

Step 2: Note that the magnitudes of the vectors are 1 and √2, respectively.

(sin x - cos x) ≤ √(1^2 (-1)^2) √2

Step 3: The maximum value of the expression is √2, and the minimum value is -√2.

Incorporating the geometric intuition, we find that the minimum value of cos x - sin x is -√2.

Conclusion

There are multiple ways to approach the problem of finding the minimum value of cos x - sin x. Calculus, trigonometric identities, and the Cauchy–Schwarz inequality all lead to the same result: the minimum value is -√2.

Keywords

cos x - sin x, minimum value, trigonometric identities