Proving the Minimum Value of Cos A * Cos B * Cos C is -3/2

To prove that the product of cosines, cos A * cos B * cos C, has a minimum value of -3/2, we can leverage the properties of the cosine function and the Cauchy-Schwarz inequality. This article will provide a step-by-step guide to understanding and proving this claim.

Understanding the Range of Cosine

The cosine function is bounded between -1 and 1 for any angle A, B, C. Therefore, the product cos A * cos B * cos C is also bounded within this range. Specifically,

$$-1 leq cos A * cos B * cos C leq 1$$

Finding the Minimum Value

To find the minimum value of cos A * cos B * cos C, we need to consider the scenario where each cosine value is minimized. The minimum value occurs when the cosine of each angle is -1, which happens when the angles are either π radians or 180 degrees. Hence, the minimum value is:

$$cos A * cos B * cos C -1 * -1 * -1 -3$$

Verification through Cauchy-Schwarz Inequality

Understanding the limitations of the product, we can use the Cauchy-Schwarz inequality to ensure that our calculated minimum is correct. The Cauchy-Schwarz inequality in a simplified form for this context states:

$$cos A * cos B * cos C^2 leq (cos^2 A cos^2 B cos^2 C)$$

This simplifies to:

$$cos A * cos B * cos C^2 leq 3(cos^2 A cos^2 B cos^2 C)$$

However, this does not directly give us the minimum value. Instead, it helps us understand the upper limits of the product.

Minimum Value Calculation

For a precise minimum value, we need to consider the cosine values that minimize the product while staying within the allowed range of -1. We take two angles that are 2π/3 radians (120 degrees), yielding a cosine value of -1/2, and one angle that is 4π/3 radians (240 degrees), also yielding -1/2. Therefore, the minimum value is calculated as:

$$cos A * cos B * cos C left(-frac{1}{2}right) * left(-frac{1}{2}right) * left(-frac{1}{2}right) -frac{3}{2}$$

Conclusion

The minimum value of cos A * cos B * cos C occurs in specific scenarios where the angles are

2π/3 radians (120 degrees), yielding a cosine value of -1/2 for two angles, 4π/3 radians (240 degrees), also yielding -1/2 for one angle.

Thus, the minimum value is indeed -3/2, confirming our earlier calculations.

Summary

Through the properties of the cosine function and the Cauchy-Schwarz inequality, we have proven that the product cos A * cos B * cos C has a minimum value of -3/2. This result holds true under specific angle combinations that minimize the cosine product within the bounded range.