Finding the Minimum Value of Cos Theta - Sin Theta

Understanding the minimum value of expressions such as ( cos theta - sin theta ) is essential in advanced mathematics and several practical applications. This article delves into the detailed process of identifying the minimum value, using both algebraic and calculus-based methods.

Algebraic Transformation

Let's start by transforming the original expression ( cos theta - sin theta ) into a more manageable form. We begin by expressing the given function in a different form:

First, we can rewrite it as:

cos theta - sin theta  sqrt{2} left( frac{1}{sqrt{2}} cos theta - frac{1}{sqrt{2}} sin theta right)

Recognizing that ( frac{1}{sqrt{2}} ) is both ( cos frac{pi}{4} ) and ( sin frac{pi}{4} ), we can use the angle addition formula:

cos theta - sin theta  sqrt{2} left( cos theta cos frac{pi}{4} - sin theta sin frac{pi}{4} right)

Applying the angle addition formula ( cos(A B) cos A cos B - sin A sin B ), we get:

cos theta - sin theta  sqrt{2} cos left( theta - frac{pi}{4} right)

Minimum Value Analysis

The expression ( sqrt{2} cos left( theta - frac{pi}{4} right) ) now lies within the range of the cosine function, which is ([-1, 1]). Therefore, the range of ( sqrt{2} cos left( theta - frac{pi}{4} right) ) is ([- sqrt{2}, sqrt{2}]).

The minimum value occurs when the cosine function is at its lowest, which is (-1). Multiplying by ( sqrt{2} ), the minimum value of ( cos theta - sin theta ) becomes:

(-sqrt{2})

Calculus Approach

An alternative method to find the minimum value is using calculus. Let's define ( y cos theta - sin theta ).

dy/dθ  - sin theta - cos theta

To find the critical points, set ( frac{dy}{dtheta} 0 ):

- sin theta - cos theta  0

Solving this equation, we get:

sin theta  -cos theta

Dividing by ( cos theta ), we obtain:

tan theta  -1

The value of ( theta ) that satisfies this equation is ( theta 135^circ ) (or ( frac{3pi}{4} ) radians).

Next, we evaluate the second derivative to determine if this point is a minimum:

d^2y/dtheta^2  -cos theta sin theta

Evaluating at ( theta frac{3pi}{4} ):

d^2y/dtheta^2  -cos frac{3pi}{4} sin frac{3pi}{4}  -left(-frac{1}{sqrt{2}}right) left(frac{1}{sqrt{2}}right)  frac{1}{2}

Since the second derivative is positive, this point is a local minimum.

Evaluating ( y ) at ( theta frac{3pi}{4} ):

y  cos frac{3pi}{4} - sin frac{3pi}{4}  -frac{1}{sqrt{2}} - frac{1}{sqrt{2}}  -sqrt{2}

Conclusion

To summarize, the minimum value of ( cos theta - sin theta ) is ( -sqrt{2} ), which occurs at ( theta 135^circ ) or ( theta frac{3pi}{4} ) radians.

Additional Notes

Understanding the behavior of trigonometric expressions is crucial in various mathematical and engineering applications. This article has provided a comprehensive approach to solving this problem using both algebraic and calculus-based methods, ensuring a deep understanding of the topic.