Problem 68 - Precalculus : Building Concepts and Connections

Question

The form of a light wave is given by the function \(f(x)=3 \cos \left(5 x-\frac{\pi}{2}\right)+4\) What are the minimum and maximum values of this function, and what is the smallest positive value of \(x\) at which the function attains its minimum value?

Step-by-Step Solution

Answer

The minimum and maximum values of the given function are 1 and 7, respectively. The smallest positive value of \(x\) at which the function attains its minimum value is \(\frac{\pi}{5}\).

Step 1: Identify \(A\), \(B\), \(C\), and \(D\) in the given function

In the given function \(f(x)=3 \cos \left(5 x-\frac{\pi}{2}\right)+4\), we can identify \(A\), \(B\), \(C\), and \(D\) as follows: \(A = 3\), \(B = 5\), \(C = -\frac{\pi}{2}\), and \(D = 4\).

Step 2: Find the minimum and maximum values

The minimum and maximum values of a cosine function of this form are given by \(D \pm A\). Hence, Minimum value = \(D - A = 4 - 3 = 1\) Maximum value = \(D + A = 4 + 3 = 7\)

Step 3: Find 'x' when the function is at its minimum

The function will reach its minimum value when the cosine function is at its lowest, which is -1, i.e. when \[\cos\left(5x - \frac{\pi}{2}\right) = -1\]. Solving the equation gives \(x = \frac{\pi + 2\pi n}{5}\) where \(n\) is an integer. The smallest positive value of \(x\) is obtained by putting \(n = 0\), and hence \(x = \frac{\pi}{5}.\)

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions, like the cosine function in the exercise, are fundamental in mathematics, particularly in studying periodic phenomena such as sound waves, light waves, and cycles of temperature changes. These functions represent the relationship between the angles of a triangle and the lengths of its sides in a right-angled triangle. They also extend these relationships to all real numbers using the unit circle.

At the heart of these functions are the sine, cosine, and tangent, each with a unique graph that oscillates between a set of maximum and minimum values also known as the peaks and troughs of the waves. The cosine function, specifically, starts from its maximum value when the angle (or the input to the function) is zero and oscillates between this maximum and its minimum as the angle increases.

Amplitude and Vertical Shift

The amplitude and vertical shift are two critical attributes that alter the appearance of a trigonometric graph. The amplitude refers to the height of the wave from the center line to a peak or trough. In other words, it is half the distance between the maximum and minimum values a trigonometric function can take. In the problem at hand, the amplitude is represented by the coefficient of the cosine function, which is 3.

The vertical shift, on the other hand, indicates how far the entire graph of the function is moved up or down along the y-axis. This shift doesn’t change the shape or the period of the function but merely relocates the graph vertically. In the given function, the vertical shift is indicated by the value added to the cosine function, which is +4, shifting the entire wave up by this value.

Period of a Cosine Function

The period of a cosine function is the length of one complete cycle of the wave — the distance on the x-axis before the function begins to repeat its pattern. This property is essential when investigating wave properties like the light wave represented in the exercise.

Generally, the period of a basic cosine function,\( \text{cos}(x) \), is \( 2\text{π} \). However, when the cosine function is multiplied by a constant, B, as in \( \text{cos}(Bx) \), the period is changed. The new period can be calculated using the formula \( \frac{2\text{π}}{|B|} \). In the given function, this constant is 5, so the period of the light wave is \( \frac{2\text{π}}{5} \), meaning after this interval, the oscillations of the light wave repeat their pattern.