Problem 30 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\) $$ y=3 \sin \frac{\pi}{2}(x-2) $$

Step-by-Step Solution

Answer

The graph of \( y = \cot(\theta) \) from \(\theta=0\) to \(\theta=2\pi\) will have vertical asymptotes at \(\theta = 0\) and \(\theta = \pi\), crossing at \(\theta=\pi/2\) and \(\theta=3\pi/2\), forming two distinctive shapes within the interval. This pattern repeats itself every \( \pi \) units.

Step 1: Understand Cotangent Function

The cotangent function, \(y = \cot(\theta)\), is the reciprocal of the tangent function. It has a period of \( \pi \) and is undefined for \( \theta = n\pi \) where n is an integer.

Step 2: Define the Interval

The interval given is from 0 to \(2\pi\). So plot points from 0, increasing each time by \(\pi/2\), up to \(2\pi\). Here, \(\theta = 0\) and \(\theta = \pi\) are vertical asymptotes, because cotangent function is not defined at these values.

Step 3: Calculate Y Values

Observe that at \(\theta=\pi/2\), the function will be 0 as \(\cot(\pi/2) = 0\). At \(\theta = 3\pi/2\), again \(\cot(3\pi/2) = 0\).

Step 4: Plot Each Point

Plot each of these points on the graph beginning from \(\theta = 0\) to \(\theta = 2\pi\). Draw smooth curve connecting the points and respect the asymptotes where the function is undefined.

Step 5: Repeat Pattern

You have now fully graphed one repetition of the function. Since the cotangent function has a period of \( \pi \), the shape you just drew will repeat every \( \pi \) steps.

Key Concepts

Cotangent Function

The cotangent function, denoted as \(y = \cot(\theta)\), plays an essential role in trigonometry. It is the reciprocal of the tangent function, which means \( \cot(\theta) = \frac{1}{\tan(\theta)}\). Therefore, understanding the behavior of the cotangent function is pivotal in grasping more complex trigonometric concepts.

A key feature of the cotangent function is that it is undefined at angles where \(\tan(\theta) = 0\). This occurs when \(\theta\) is a multiple of \(\pi\), such as \(0\), \(\pi\), etc. In these cases, the denominator of the reciprocal is zero, leading to an undefined value.

When you graph \(y = \cot(\theta)\), you will notice the curve has a distinct pattern. Between its asymptotes, it starts from a high positive value, passes through zero, and then descends towards a high negative value. The graph repeats this pattern due to the regular periodicity inherent in trigonometric functions.

Trigonometric Asymptotes

Trigonometric asymptotes are vertical lines on a graph where the function is undefined. In the context of the cotangent function, these asymptotes occur at specific intervals.

For \(y = \cot(\theta)\), asymptotes appear wherever \(\theta = n\pi\), where \(n\) is an integer. This results from the function’s reciprocation property with tangent, which has zero values at multiples of \(\pi\).

Identifying and respecting these asymptotes while graphing is crucial. They indicate where the function shoots off to infinity and can dramatically switch from positive to negative or vice versa.

• In our interval from \(0\) to \(2\pi\), key asymptotes are observed at \(\theta = 0\) and \(\theta = \pi\).
• These asymptotes divide the function into segments, creating a consistent pattern of behavior across each period.

Understanding these asymptotic lines helps in drawing accurate graphs and is also helpful in solving complex trigonometric equations.

Period of Trigonometric Functions

The period of a trigonometric function describes how often it repeats its pattern. For the cotangent function, \(y = \cot(\theta)\), the period is \(\pi\).

This means that every \(\pi\) units along the \(\theta\)-axis, the function’s graph will repeat the entire pattern.

To visualize this, consider how the cotangent graph looks from \(0\) to \(\pi\). It will show the same sequence of rising, zero, and falling like a mirror reflection between successive asymptotes.

• Graphing from \(\theta = 0\) to \(\theta = 2\pi\) will showcase two complete cycles of the cotangent function.
• At \(\pi/2\), \(3\pi/2\), etc., the function intersects the horizontal axis, which are points where \(y = 0\).

Knowing the period is critical when tackling problems involving transformations and phase shifts in trigonometric functions.