Question

ERROR ANALYSIS Describe and correct the error in finding the period of the function \(y=\cot 3 x\).

Short answer

The correct period of the function \(y=\cot 3 x\) is \(\pi /3\). The error comes from not correctly applying the formula for the period of the function \(y=\cot kx\), which is \(\pi / |k|\).

Step-by-step solution

Error Identification

The first step in error analysis is identifying the error. The error might come from incorrectly applying the formula for the period of the function \(y= \cot kx\). The correct formula is period = \(\pi / |k|\).

Applying Correct Formula

Apply the correct formula for finding the period of the function \(y= \cot 3x\). Here, the value of k is 3. So applying the formula, the period would be \(\pi / 3\). This is because, for the cotangent function, period is scaled by a factor of \(1/k\). This scaling factor reduces the length of the period by the same amount.

Verifying Correctness

To ensure the answer is correct, verify that the period of the function \(y= \cot 3x\) is \(\pi / 3\). This is done by checking whether one complete cotangent wave is completed in the interval \(0 \leq x < \pi /3\).

Key concepts

Error Analysis

Error analysis is a critical process in mathematics, where you identify and correct mistakes made during problem-solving or calculations. In the context of trigonometric functions, like the cotangent, this often involves verifying whether the correct formula is used to determine specific properties of the function.

• Identify the error: This is the initial step where you closely examine the problem to find any mistakes.
• Apply the correct method: Once the error is spotted, apply the appropriate formula or methodology to rectify it.
• Verify the correction: Finally, make sure the correction is accurate by double-checking against known values or properties.

Using these steps can significantly improve your problem-solving skills and ensure accurate results. In the given example, recognizing an incorrect application of the period formula for the cotangent function was key in correcting the error.

Cotangent Function

The cotangent function, denoted as \(y = \cot x\), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function, meaning \(\cot x = \frac{1}{\tan x}\) or \(\cot x = \frac{\cos x}{\sin x}\). Understanding the behavior of the cotangent function involves several aspects:

• Reciprocal nature: Since it is derived from the cosine and sine functions, its values will be undefined wherever the sine function is zero.
• Graph characteristics: The graph of the cotangent function shows that it has vertical asymptotes whenever the sine is zero, such as \(x = n\pi\) where \(n\) is an integer.
• Periodicity: The graph repeats after its period, which is \(\pi\) in its basic form, but can be altered by transformation.

An essential property of the cotangent graph is its repetition over intervals of \(\pi\), offering a regular pattern of ascent and descent. These properties make the cotangent function a useful tool in various mathematical applications.

Period of Function

In trigonometry, the period of a function is the interval after which the function values repeat in a regular pattern. For the cotangent function \(y = \cot x\), its basic period is \(\pi\). When modifications are made, such as scaling by a constant factor, the formula for the period becomes crucial.The formula to find the period for the cotangent function \(y = \cot kx\) is \(\text{period} = \frac{\pi}{|k|}\).

• This formula indicates how the frequency of the function changes, making the graph compress or stretch horizontally.
• A larger \(k\) value results in a smaller period, leading to more cycles within the same length of the x-axis.
• When analyzing functions like \(y = \cot 3x\), the period becomes \(\frac{\pi}{3}\), demonstrating three times the frequency of the standard cotangent.

These simple calculations are vital in predicting the behavior and appearance of trigonometric graphs, especially when applied in fields that require periodic analysis, such as engineering and physics.