10. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

10. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions: Videos & Practice Problems

Topic summary

The tangent function graph is derived from the identity $ \tan(x) = \frac{\sin(x)}{\cos(x)} $, featuring asymptotes at odd multiples of $ \frac{\pi}{2} $. Its period is $ \frac{\pi}{b} $, differing from sine and cosine's $ \frac{2\pi}{b} $. The cotangent graph, being the reciprocal of tangent, has asymptotes at integer multiples of $ \pi $ and mirrors the tangent's behavior but inverted. Both graphs repeat every $ \pi $ units, emphasizing their interconnectedness in trigonometric functions.

concept

Introduction to Tangent Graph

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Introduction to Tangent Graph Video Summary

In exploring the graph of the tangent function, it's essential to understand its relationship with sine and cosine. The tangent of an angle can be expressed as the ratio of sine to cosine, represented mathematically as:

\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]

Starting from the origin, when $ x = 0 $, the tangent value is:

\[ \tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0 \]

As we move to $ x = \frac{\pi}{4} $, we find:

\[ \tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \]

Conversely, at $ x = -\frac{\pi}{4} $, the tangent value is:

\[ \tan\left(-\frac{\pi}{4}\right) = \frac{\sin\left(-\frac{\pi}{4}\right)}{\cos\left(-\frac{\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 \]

At $ x = \frac{\pi}{2} $ and $ x = -\frac{\pi}{2} $, the tangent function becomes undefined because the cosine value is zero, leading to vertical asymptotes at these points. The graph of the tangent function thus approaches infinity at these asymptotes, which occur at odd multiples of $ \frac{\pi}{2} $, specifically at:

\[ x = \frac{\pi}{2}, \, -\frac{\pi}{2}, \, \frac{3\pi}{2}, \, -\frac{3\pi}{2} \]

Unlike the secant function, which has a period of $ 2\pi $, the tangent function has a period of $ \pi $. This means the graph repeats every $ \pi $ units along the x-axis. The period can be calculated using the formula:

\[ \text{Period} = \frac{\pi}{b} \]

For example, if we consider the function $ y = \tan\left(\frac{\pi}{2} x\right) $, the coefficient $ b $ is $ \frac{\pi}{2} $. Thus, the period is:

\[ \text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2 \]

This indicates that the tangent function will repeat its pattern every 2 units. To graph this function, we can place asymptotes at $ x = -1 $ and $ x = 1 $, and then extend the graph to the left and right, repeating the curve between these asymptotes. The resulting graph will show the characteristic shape of the tangent function, curving upwards to the right and downwards to the left of each asymptote.

Understanding these properties of the tangent function, including its asymptotes and periodicity, is crucial for accurately graphing and analyzing its behavior in various mathematical contexts.

Problem

Below is a graph of the function $y=\tan\left(bx\right)$ . Determine the value of b.

$b=\frac14$

$b=\pi$

$b=2$

$b=\frac12$

example

Example 1

Video duration:

Example 1 Video Summary

To graph the function $ y = \frac{1}{2} \tan\left(x - \frac{\pi}{2}\right) $, we start by analyzing the basic function $ y = \frac{1}{2} \tan(x) $. The tangent function has vertical asymptotes at odd multiples of $ \frac{\pi}{2} $, specifically at $ x = -\frac{\pi}{2} $ and $ x = \frac{\pi}{2} $. The graph of the tangent function typically passes through the origin and has points at $ \left(\frac{\pi}{4}, 1\right) $ and $ \left(-\frac{\pi}{4}, -1\right) $.

When we introduce the coefficient $ \frac{1}{2} $, it affects the amplitude of the graph, reducing the height of the tangent function. Consequently, the points shift to $ \left(\frac{\pi}{4}, \frac{1}{2}\right) $ and $ \left(-\frac{\pi}{4}, -\frac{1}{2}\right) $. Thus, the graph of $ y = \frac{1}{2} \tan(x) $ appears shorter than the standard tangent graph.

Next, we incorporate the horizontal shift caused by the term $ -\frac{\pi}{2} $. The expression $ x - \frac{\pi}{2} $ indicates a shift to the right by $ \frac{\pi}{2} $ units. To determine the new positions of the asymptotes, we take the original asymptotes and shift them accordingly. The asymptote at $ x = \frac{\pi}{2} $ moves to $ x = \pi $, and the asymptote at $ x = -\frac{\pi}{2} $ shifts to $ x = 0 $. Additionally, we can find another asymptote at $ x = -\pi $ by continuing the pattern of the tangent function.

With these adjustments, the graph will repeat every $ \pi $ units, maintaining the same shape as the original tangent graph but with the new asymptotes and points. The final graph of $ y = \frac{1}{2} \tan\left(x - \frac{\pi}{2}\right) $ will exhibit the characteristic up-and-down behavior of the tangent function, with the specified transformations applied.

concept

Introduction to Cotangent Graph

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Introduction to Cotangent Graph Video Summary

Understanding the cotangent graph is essential for mastering trigonometric functions, especially since it is closely related to the tangent graph. The cotangent function, defined as the reciprocal of the tangent function, can be graphed by identifying its asymptotes and the behavior of its curves.

To begin, recall that the cotangent function is expressed as:

\[ \cot(x) = \frac{1}{\tan(x)} \]

This relationship indicates that wherever the tangent function is zero, the cotangent function will have vertical asymptotes. For the tangent graph, these points occur at odd multiples of $\frac{\pi}{2}$, specifically at $-\frac{3\pi}{2}$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, and $\frac{3\pi}{2}$. Consequently, the cotangent graph will have vertical asymptotes at integer multiples of $\pi$, such as $-\pi$, $0$, and $\pi$.

The cotangent graph exhibits a repeating pattern, similar to the tangent graph, with a period of $\pi$. The period of a trigonometric function can be calculated using the formula:

\[ \text{Period} = \frac{\pi}{b} \]

In this case, if the coefficient $b$ in front of $x$ is $1$, the period remains $\pi$. This means that the cotangent function will repeat its values every $\pi$ units along the x-axis.

When sketching the cotangent graph, the curves will descend from the top left to the bottom right between the asymptotes. For example, between the asymptotes at $-\pi$ and $0$, the graph will start from positive infinity and approach zero, then continue from zero to negative infinity as it approaches the next asymptote at $0$. This behavior is mirrored in each subsequent interval of $\pi$.

To illustrate this, if we consider a cotangent function with a period of $1$, the asymptotes would be located at every integer value along the x-axis. The graph would then display curves that descend between these asymptotes, repeating the pattern indefinitely.

In summary, the cotangent graph is characterized by its vertical asymptotes at integer multiples of $\pi$, a period of $\pi$, and curves that descend from left to right. Understanding these properties allows for accurate sketching and analysis of the cotangent function in relation to other trigonometric functions.

Problem

Below is a graph of the function $y=\cot\left(bx+\frac{\pi}{2}\right)$ . Determine the value of b.

$b=\frac14$

$b=1$

$b=2$

$b=\frac12$

example

Example 1

Video duration:

Example 1 Video Summary

To graph the function $ y = -2 \cot\left(\frac{1}{4}x\right) $, we need to analyze several components that affect its shape and behavior. The negative sign indicates that the graph will be reflected over the x-axis, while the coefficient of 2 will vertically stretch the graph, altering its amplitude. Additionally, the factor of $ \frac{1}{4} $ inside the cotangent function will affect the period of the graph.

Starting with the basic cotangent function, the period is determined by the formula $ \frac{\pi}{b} $, where $ b $ is the coefficient of $ x $. In this case, $ b = \frac{1}{4} $, so the period becomes:

\[ \text{Period} = \frac{\pi}{\frac{1}{4}} = 4\pi \]

This means the cotangent function will repeat every $ 4\pi $ units. The asymptotes of the cotangent function occur at $ x = 0 $, $ x = 4\pi $, and $ x = 8\pi $. The cotangent graph typically approaches infinity as it nears the asymptotes, decreasing from the left and increasing from the right, with a zero crossing at the midpoint between asymptotes, which is at $ 2\pi $.

Next, we identify key points. The cotangent function has values of 1 and -1 at $ x = \pi $ and $ x = 3\pi $, respectively. However, since we are dealing with $ -2 \cot\left(\frac{1}{4}x\right) $, we need to adjust these outputs. The vertical stretch means that at $ x = \pi $, the output will be $ -2 $ instead of $ -1 $, and at $ x = 3\pi $, the output will be $ 2 $ instead of $ 1 $.

Thus, the adjusted points are:

At $ x = \pi $, $ y = -2 $
At $ x = 3\pi $, $ y = 2 $

With these transformations, the graph of $ y = -2 \cot\left(\frac{1}{4}x\right) $ will start from the lower left, rise to cross the x-axis at $ x = 2\pi $, and continue to the upper right, reflecting the periodic nature of the cotangent function. The graph will repeat this pattern between each set of asymptotes, creating a continuous wave-like structure.

In summary, the function $ y = -2 \cot\left(\frac{1}{4}x\right) $ features a period of $ 4\pi $, vertical stretching by a factor of 2, and reflection over the x-axis, resulting in a unique graph that captures these transformations effectively.

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Here’s what students ask on this topic:

What are the key characteristics of the tangent function graph?

The tangent function graph, derived from the identity $\tan (x) = \frac{\sin}{(}$ , features several key characteristics. It has vertical asymptotes at odd multiples of $\frac{π}{/}$ (e.g., $\frac{π}{/}$ , $\frac{3π}{/}$ ). The period of the tangent function is $\frac{π}{/}$ , where $b$ is the coefficient of $x$ . The graph repeats every $π$ units and passes through the origin, with the curve moving from the bottom left to the top right between asymptotes.

How do you graph the cotangent function?

To graph the cotangent function, recall that $\cot (x) = \frac{1}{/}$ . Start by identifying the asymptotes, which occur at integer multiples of $π$ (e.g., $0$ , $π$ , $2π$ ). The period of the cotangent function is $\frac{π}{/}$ , where $b$ is the coefficient of $x$ . The graph repeats every $π$ units. The curves of the cotangent graph move from the top left to the bottom right between asymptotes, opposite to the tangent graph.

What is the period of the tangent function?

The period of the tangent function is given by $\frac{π}{/}$ , where $b$ is the coefficient of $x$ in the function $\tan (bx)$ . This differs from the sine and cosine functions, which have a period of $\frac{2π}{/}$ . For example, if the function is $\tan (\frac{π}{/}$ , the period would be $\frac{π}{/}$ , which simplifies to 2.

How do the asymptotes of the tangent and cotangent functions differ?

The asymptotes of the tangent and cotangent functions differ in their locations on the x-axis. For the tangent function, asymptotes occur at odd multiples of $\frac{π}{/}$ (e.g., $\frac{π}{/}$ , $\frac{3π}{/}$ ). For the cotangent function, asymptotes occur at integer multiples of $π$ (e.g., $0$ , $π$ , $2π$ ). This difference is due to the reciprocal relationship between the tangent and cotangent functions.

What is the relationship between the tangent and cotangent graphs?

The tangent and cotangent graphs are closely related but have distinct characteristics. The tangent function, $\tan (x)$ , has vertical asymptotes at odd multiples of $\frac{π}{/}$ and repeats every $π$ units. The cotangent function, $\cot (x)$ , being the reciprocal of tangent, has vertical asymptotes at integer multiples of $π$ and also repeats every $π$ units. The cotangent graph is essentially the tangent graph flipped upside down, with curves moving from the top left to the bottom right between asymptotes.