Problem 12 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=2 \cot x$$

Step-by-Step Solution

Answer

The period of \(y = 2 \cot x\) is \(\pi\).

Step 1: Identifying the Standard Form

The given function is \(y = 2 \cot x\). The standard form of a cotangent function is \(y = a \cot(bx + c) + d\). Here, \(a = 2\), \(b = 1\), \(c = 0\), and \(d = 0\).

Step 2: Finding the Period of the Function

The period of a cotangent function \(y = \cot(bx)\) is given by \(\frac{\pi}{b}\). In this case, \(b = 1\), so the period is \(\frac{\pi}{1} = \pi\).

Step 3: Graphing the Function

To graph \(y = 2 \cot x\), start by plotting the basic shape of \(\cot x\), which has vertical asymptotes at multiples of \(\pi\) and crosses the x-axis at multiples of \(\frac{\pi}{2}\). The amplitude \(a = 2\) stretches the graph vertically by a factor of 2.The asymptotes will remain at \(0, \pi, 2\pi, \ldots\) and the zeros at \(\frac{\pi}{2}, \frac{3\pi}{2}, \ldots\).

Step 4: Check Graph Characteristics

Ensure that between each pair of asymptotes (e.g., 0 and \(\pi\), \(\pi\) and \(2\pi\)), the graph decreases from positive infinity to negative infinity, crossing the x-axis at half the interval. This reflects the cotangent function properties with a period of \(\pi\) and scaled by 2.

Key Concepts

Cotangent Function

The cotangent function is one of the basic trigonometric functions. It is the reciprocal of the tangent function. When you hear cotangent, think about how it relates to standard functions like sine and cosine too. The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, it can be expressed as:\[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\]

• Reciprocal Nature: Since cotangent is the reciprocal of tangent, it becomes undefined when the sine value is zero, which happens at points where \( x = n\pi \) where \( n \) is an integer.
• Basic Graph Features: Standard cotangent graph has vertical asymptotes at these points because it tends to positive or negative infinity depending on the approach to the asymptote.

Understanding cotangent well, you'll better understand other trigonometric identities and their relationships. Remember that the transformations of the cotangent function, like scaling and translation, alter its graph. This knowledge helps you graph these functions easily.

Period of Trigonometric Functions

The period of a function refers to how often the function repeats itself. For a trigonometric function, this is an important characteristic since they are inherently periodic. Periodicity means that there is a regular and repeating pattern, which makes these functions very predictable.For the general form of a cotangent function, \( y = a \cot(bx + c) + d \), the period is directly influenced by the value of \( b \). The period of a basic cotangent function \( \cot(x) \) is \( \pi \), but it changes with a coefficient \( b \) in front of \( x \).

• Formula to Find Period: For the function \( \cot(bx) \), use the formula: \( \text{Period} = \frac{\pi}{b} \). If \( b = 1 \), the period remains \( \pi \).
• Applications: Knowing the period allows you to precisely graph and predict the behavior of the function over intervals. For semesters like exams, this deeper understanding contributes to a number of problem-solving strategies.

In the case of \( y = 2 \cot x \), with \( b = 1 \) and no horizontal shifts, the period is \( \pi\), ensuring each cycle of the graph behaves in a familiar and repeated manner.

Graphing Trigonometric Functions

Successfully graphing trigonometric functions involves considering their characteristics, such as period, amplitude, and phase shifts. For the function \( y = 2 \cot x \), it is crucial to track these attributes to map the graph accurately.

• Vertical Stretch: The coefficient, here \( a = 2 \), changes the vertical stretch of the graph. Since it multiplies \( \cot(x) \), every value of \( \cot(x) \) becomes 2 times as tall of its original, resulting in a steeper curve.
• Identifying Asymptotes: Vertical asymptotes are determined by the function becoming undefined. In \( y = 2 \cot x \), they remain at special values \( 0, \pi, 2\pi, \ldots\).
• Plotting Key Points: Recognize that between asymptotes, the cotangent function smoothly transitions from positive infinity to negative infinity, crossing the x-axis at \( \frac{\pi}{2}, \frac{3\pi}{2},\ldots \).
• Consistency in Cycles: Due to periodicity, replicating cycles of the graph accurately after one interval (\pi) helps in sketching out the function over a longer domain easily.

By emphasizing the function's symmetry and repeating behaviors, learners easily manage the graphing of even complex trigonometric forms. This skill is critical for visualizing solutions and verifying analytical calculations involving trigonometric functions.