In the quest to find the minimum value of the expression 4 - 3cos x / 3sin x, we must consider the complexities involved without relying on calculus or graphing. This problem requires deep understanding and manipulation of trigonometric identities. Let's delve into the process step by step.

Understanding the Expression and Function Behavior

The given function can be rewritten as:

[ f(x) frac{4 - 3cos x}{3sin x} ]

Given that (sin x) is in the denominator, this function repeats every (2pi) units and has vertical asymptotes at every multiple of (pi). The function is negative from (pi) to (2pi), then increasing to a relative maximum and decreasing thereafter.

To locate the relative minimum of (f(x)) between (0) and (pi), we need to consider the behavior of the numerator and the denominator.

Rewriting the Expression

The function can be rewritten as:

[ f(x) frac{4}{3} cdot csc x cdot cot x ]

Both (csc x) and (cot x) have specific behaviors as (x) approaches certain values:

(csc x) approaches (infty) as (x) approaches (0) from the right. (cot x) approaches (-infty) as (x) approaches (pi) from the left.

Considering these behaviors, a good guess for the relative minimum would be (frac{3pi}{4}).

Verifying the Minimum Value

Let's verify the value of (f(x)) at (frac{3pi}{4}) and compare it with nearby values:

(fleft(frac{pi}{2}right) frac{4}{3} cdot csc left(frac{pi}{2}right) cdot cot left(frac{pi}{2}right) frac{4}{3} cdot 1 cdot 0 0)

(fleft(frac{pi}{3}right) frac{4}{3} cdot csc left(frac{pi}{3}right) cdot cot left(frac{pi}{3}right) frac{4}{3} cdot frac{2}{sqrt{3}} cdot frac{1}{sqrt{3}} frac{4}{3} cdot frac{2}{3} frac{8}{9})

(fleft(frac{pi}{4}right) frac{4}{3} cdot csc left(frac{pi}{4}right) cdot cot left(frac{pi}{4}right) frac{4}{3} cdot sqrt{2} cdot 1 frac{4sqrt{2}}{3})

(fleft(frac{pi}{6}right) frac{4}{3} cdot csc left(frac{pi}{6}right) cdot cot left(frac{pi}{6}right) frac{4}{3} cdot 2 cdot sqrt{3} frac{8sqrt{3}}{3})

From these calculations, the minimum value around (frac{3pi}{4}) seems to be slightly less than this value. This confirms our initial guess.

Alternative Approach Without Calculus

Considering the expression (4 - 3cos x / 3sin x), we can analyze the behavior of the numerator and the denominator:

At (x frac{pi}{2}), the value of the expression is 0. At (x frac{pi}{3}), the value is more than 0. At (x frac{pi}{4}), the value is even larger. At (x frac{pi}{6}), the value is still larger.

By checking more values, we can conclude that the minimum value of the expression is between these points, confirming our earlier guess.

Conclusion

Through careful analysis and the use of trigonometric identities, we have identified that the minimum value of the expression (4 - 3cos x / 3sin x) between (0) and (pi) is indeed around (frac{3pi}{4}). This method avoids the use of calculus and provides a practical way to find the minimum value of such a trigonometric function.

Keywords: minimum value, trigonometric functions, without calculus