Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\cot \left(x+\frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

Period: \(\pi\), Phase shift: \(-\frac{\pi}{3}\), Range: \((-\infty, \infty)\).

Step 1: Identify the function

The given function is \[y = \cot \left(x + \frac{\pi}{3}\right)\]This is a cotangent function which has specific characteristics that we need to analyze.

Step 2: Calculate the period

For the cotangent function, the period is given by \[\pi\div |B|\]where \(B\) is the coefficient of \(x\). Here, \(B = 1\). Therefore, the period of the given function is \[\pi\div 1 = \pi.\]

Step 3: Determine the phase shift

The phase shift of the function is determined by \[-\frac{C}{B}\]where \(C\) is the constant added to \(x\) inside the function. For this function, \(C = \frac{\pi}{3}\). Since \(B = 1\), the phase shift is \[-\frac{\pi/3}{1} = -\frac{\pi}{3}.\]This indicates a phase shift to the left by \(\frac{\pi}{3}\).

Step 4: Identify the range

The range of the cotangent function is all real numbers, denoted as \[(-\infty, \infty)\].Therefore, the range for this function is \((-\infty, \infty)\).

Key Concepts

Cotangent Function

The cotangent function, often abbreviated as cot, is a fundamental trigonometric function. The formula for the cotangent of an angle is given by:

• cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)

The cotangent function relates to an angle in a right triangle. It is defined as the ratio between the adjacent side and the opposite side, similar to how the tangent function works but inverted.

The graph of the cotangent function differs from that of sine, cosine, and tangent due to its reciprocal nature. Notably, cotangent has vertical asymptotes and is undefined at angles where the sine is zero. Understanding the behavior of cotangent helps in grasping more complex trigonometric equations and real-world applications, like in wave oscillations.

Period

The period of a trigonometric function describes how frequently it repeats its pattern along the x-axis. For the cotangent function, the standard period is \[ \pi \].

• In the formula for a trigonometric function of the form \[ y = a \cot(bx + c) \], the period is calculated by \[ \frac{\pi}{|b|} \].

In the given function \[ y = \cot(x + \frac{\pi}{3}) \], the coefficient \[ b \] is 1. This means the period of the function is \[ \pi \].

The period is important as it tells you the length over which the cotangent function completes one full cycle on the graph. It's a key concept when examining trigonometric functions as it helps predict the function's behavior.

Phase Shift

The phase shift is a horizontal translation of the graph of the function. It indicates the starting point of the function's cycle. For a function of the form \[ y = a \cot(bx + c) \], the phase shift is calculated using this formula:

• \[ -\frac{c}{b} \]

For the given function \[ y = \cot(x + \frac{\pi}{3}) \], the value of \[ c \] is \[ \frac{\pi}{3} \], and \[ b \] is 1.

Thus, the phase shift is:

• \[ -\frac{\pi}{3} \]

This means the graph of the function is shifted to the left by \[ \frac{\pi}{3} \].
Phase shift is essential for understanding how changes within the function affect its position on the trigonometric circle. This can help address issues in both pure and applied mathematics.

Range

The range of a function refers to the set of possible output values it can produce. For the cotangent function, the range is very distinct.

• The cotangent function’s range is all real numbers: \((-\infty, \infty)\).

This wide range means that any real number can result from the cotangent function, which distinguishes it from other trigonometric functions like sine and cosine, which have limited ranges between -1 and 1.
Understanding the range is important because it tells us how the function behaves, helping us predict all possible outcomes for inputs within its domain. This broader perspective is particularly helpful when modeling real-world phenomena where outputs can span across all real numbers.