Problem 13 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=-\cot x $$

Step-by-Step Solution

Answer

The period of \( y = -\cot x \) is \( \pi \).

Step 1: Understand the Cotangent Function

The cotangent function, \( \cot x \), is defined as the reciprocal of the tangent function. In terms of sine and cosine, it can be expressed as \( \cot x = \frac{\cos x}{\sin x} \). The function \( y = -\cot x \) is simply the cotangent function reflected over the x-axis.

Step 2: Determine the Period of \( \cot x \)

The period of the cotangent function \( \cot x \) is \( \pi \). This is because within any interval of length \( \pi \), the cotangent function completes one full cycle of its behavior.

Step 3: Analyze the Reflection Effect

Reflecting the cotangent function across the x-axis to obtain \( y = -\cot x \) does not affect the period of the function. Thus, \( y = -\cot x \) still has a period of \( \pi \).

Step 4: Identify Important Features for Graphing

The cotangent function has vertical asymptotes where \( \sin x = 0 \), which occur at integer multiples of \( \pi \), such as \( x = 0, \pi, 2\pi, \ldots \). The graph of \( y = -\cot x \) will have these same asymptotes, with the key difference being the function is "flipped" over the x-axis.

Step 5: Plot the Function for One Period

To graph \( y = -\cot x \) for one period, plot the curve starting just after \( x = 0 \), descending through the origin \( (\pi/2, 0) \) and approaching the asymptote at \( x = \pi \) from above. The function has negative values in this interval because of the reflection.

Key Concepts

Periodic Functions

Periodic functions are an important concept in mathematics, particularly in the study of waves and oscillations. These functions repeat their values in regular intervals or periods. To better understand periodic functions, let's look at their key features:

• **Period**: This is the smallest positive interval over which the function repeats itself. For example, in the case of the cotangent function, its period is \( \pi \).
• **Repetitive Patterns**: A defining feature of periodic functions is that they exhibit repetitive behavior. So, when you plot a periodic function, you see the same shape again and again at regular intervals.
• **Real-world Applications**: Periodic functions model many natural phenomena such as sound waves, light waves, and tides.

Understanding periodic functions is crucial as they form the basis for trigonometric functions like sine, cosine, and cotangent, which naturally repeat due to their geometric origins in the unit circle.

Trigonometric Functions

Trigonometric functions are a class of functions that arise from the relationships between the angles and sides of triangles, specifically right-angled triangles. They are foundational in mathematics and have multiple applications in different fields.

• **Basic Trigonometric Functions**: The primary trigonometric functions are sine, cosine, and tangent, with secant, cosecant, and cotangent being their reciprocals. Each of these functions can express angles as ratios of side lengths in a right triangle.
• **Unit Circle Representation**: Trigonometric functions are effectively represented on the unit circle, where angles correspond to rotations, and function values are read directly from the circle.
• **Periodic Nature**: All trigonometric functions are periodic. For example, both sine and cosine have a period of \(2\pi\), while the tangent and cotangent functions have a period of \(\pi\).

A specific example of a trigonometric function is the cotangent, which is defined for any angle as \( \cot x = \frac{\cos x}{\sin x} \). It is essential in functions like \(y = -\cot x\), which involves reflecting the cotangent function across the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a range, showing their periodic and oscillatory behavior. The process requires understanding the function's period, amplitude, and other properties to sketch an accurate graph.

• **Period Identification**: For graphing, knowing the period is crucial, as it indicates the interval over which the function repeats. For the cotangent function, the period is \( \pi \).
• **Graph Features**: For the cotangent and its reflection \( y = -\cot x \), important features include vertical asymptotes where the function is undefined (e.g., \( x = 0, \pi, 2\pi, \ldots \)), zeros where the function crosses the x-axis, and the general shape of its oscillation between asymptotes.
• **Reflection and Transformation**: Reflection across the x-axis, as in \( y = -\cot x \), flips the graph vertically, impacting its values while maintaining other core features like period and asymptotes.

When graphing \( y = -\cot x \), one must correctly place these asymptotes and plot points to depict the function's descent and reflection accurately from \( \pi/2 \) to \( \pi \). These steps ensure the trigonometric function's graph is both functional and visual.