How to Find the Minimum Value of 3cos x 4sin x 8

When dealing with trigonometric expressions such as 3cos x 4sin x, finding the minimum value can be approached through various methods. This article will explore how to find the minimum value of 3cos x 4sin x 8 using both direct methods and optimization techniques.

Step 1: Rewriting the Trigonometric Expression

A common approach is to rewrite the expression 3cos x 4sin x in the form Rcos(x - φ) where R is the amplitude and φ is a phase shift. This is done to simplify the expression and make it easier to analyze.

Step 1.1: Determining the Amplitude R

To find R, we use the formula:

R sqrt{a^2 b^2}

Given that a 3 and b 4, we calculate:

R sqrt{3^2 4^2} sqrt{9 16} sqrt{25} 5

Step 1.2: Finding the Phase Shift φ

The phase shift φ can be determined using:

cos φ frac{a}{R} frac{3}{5}

sin φ frac{b}{R} frac{4}{5}

With R and φ determined, we can now rewrite the original expression:

3cos x 4sin x 5cos(x - φ)

Step 2: Determining the Minimum Value

The cosine function has a minimum value of -1. Therefore, the minimum value of 5cos(x - φ) is:

5(-1) -5

Adding the constant 8 to -5 gives us:

text{Minimum value} -5 8 3

Alternative Methods for Finding the Minimum Value

Method 1: Direct Substitution

Another approach is to let 3 5 sin t and 4 5 cos t. Then:

P 3cos x 4sin x 8 5sin t x 8

The minimum value of 5sin t is -5, so:

Pmin -5 8 3

Method 2: Optimization Using Derivatives

Alternative Optimization (AOD) methods involve finding the critical points of the function by differentiating it and setting the derivative to zero. For the function:

fx 3cos x 4sin x 8

we take the first derivative:

f'{x} -3sin x 4cos x

Setting f'{x} 0 gives us:

tan x frac{4}{3}

x 53° or 233°

Next, we take the second derivative:

f''{x} -3cos x 4sin x

At x 53°, f''{x} -5 (indicating a maximum), and at x 233°, f''{x} 5 (indicating a minimum). Therefore, the function is minimum at x 233°, and the value of the expression at this point is 3.

Conclusion

The minimum value of 3cos x 4sin x 8 is boxed{3}.

Using these methods, we can effectively determine the minimum value of complex trigonometric expressions. Whether through direct substitution, optimization techniques, or derivative analysis, the key steps involve rewriting the expression, identifying critical points, and determining the minimum or maximum values.