Graphs of Other Trigonometric Functions: Videos & Practice Problems

Understanding the graphs of trigonometric functions is essential, particularly for cosecant, secant, tangent, and cotangent. The cosecant and secant are reciprocals of sine and cosine, respectively, with asymptotes at points where sine or cosine equals zero. The tangent function, defined as sine over cosine, has asymptotes at odd multiples of π/2 and a period of π. The cotangent graph mirrors the tangent but is flipped, with asymptotes at integer multiples of π. Both functions share transformation rules, enhancing their graphing process.

Graphs of Secant and Cosecant Functions

In trigonometry, the cosecant and secant functions are essential extensions of the sine and cosine functions, respectively. Understanding their graphs involves recognizing that cosecant is the reciprocal of sine, expressed mathematically as csc(x) = 1/sin(x), and secant is the reciprocal of cosine, given by sec(x) = 1/cos(x). This relationship is crucial for graphing these functions effectively.

When graphing the cosecant function, it is important to note that it is undefined wherever the sine function equals zero, which occurs at integer multiples of π (i.e., 0, π, 2π, ...). At these points, vertical asymptotes are drawn, indicating that the cosecant function approaches infinity. For example, at x = 0, π, and 2π, the cosecant function is undefined. The behavior of the cosecant graph can be visualized as "smiley" and "frowny" faces, where the peaks correspond to the maximum values of the sine function and the valleys correspond to the minimum values.

Similarly, the secant function is undefined wherever the cosine function equals zero, which occurs at odd multiples of π/2 (i.e., π/2, 3π/2, ...). As with cosecant, vertical asymptotes are placed at these points. The secant graph also exhibits a similar pattern of peaks and valleys, but the locations of the asymptotes differ from those of the cosecant function.

Both cosecant and secant functions can be transformed using the same rules applied to sine and cosine functions, such as stretching and shifting. For instance, to graph y = csc(2x), one would first graph y = sin(2x) to determine the key points and asymptotes. The period of the sine function is calculated using the formula Period = 2π/b, where b is the coefficient of x. In this case, with b = 2, the period becomes π.

After identifying the sine graph's peaks and zeros, the cosecant graph can be constructed by placing asymptotes at the zeros of the sine function and drawing the characteristic "smiley" and "frowny" shapes between these asymptotes. This method allows for a clear understanding of the behavior of both cosecant and secant functions, reinforcing their relationship with sine and cosine.

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

• A. $b=2$
• B. $b=4$
• C. $b=\frac{\pi}{2}$
• D. $b=\pi$

Answer: (A)

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

• A. $b=\frac12$
• B. $b=3$
• C. $b=\frac43$
• D. $b=\frac{3\pi}{4}$

Answer: (C)

Introduction to Tangent Graph

In exploring the graph of the tangent function, it's essential to build on the foundational knowledge of sine and cosine functions. The tangent function can be defined using the identity:

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

This relationship allows us to graph the tangent function by evaluating the sine and cosine values at various points. Starting at \(x = 0\), we find:

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

At \(x = \frac{\pi}{4}\), we have:

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

Conversely, at \(x = -\frac{\pi}{4}\), the 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 due to division by zero:

\[ \tan\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} = \frac{1}{0} \quad \text{(undefined)} \]

This results in vertical asymptotes at these points, indicating that the function approaches infinity. The asymptotes for the tangent function occur at odd multiples of \(\frac{\pi}{2}\), specifically at:

\[ x = \frac{\pi}{2} + k\pi \quad (k \in \mathbb{Z}) \]

Thus, the tangent graph has a repeating pattern with a period of \(\pi\), contrasting with the period of \(2\pi\) seen in sine and cosine functions. The formula for the period of the tangent function is given by:

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

where \(b\) is the coefficient of \(x\) in the function. For example, in the function:

\[ y = \tan\left(\frac{\pi}{2} x\right) \]

the period can be calculated as:

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

This indicates that the tangent function will repeat every 2 units along the x-axis. When graphing, it is crucial to mark the asymptotes and the repeating curves accurately. For instance, if the first asymptote is at \(x = -1\), the next will be at \(x = 1\), and the curve will repeat between these points.

In summary, understanding the tangent function involves recognizing its relationship to sine and cosine, identifying asymptotes, and applying the correct period formula. This foundational knowledge allows for accurate graphing and analysis of the tangent function's behavior.

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

• A. $b=\frac14$
• B. $b=\pi$
• C. $b=2$
• D. $b=\frac12$

Answer: (D)

Introduction to Tangent Graph Example 1

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

Next, we introduce the coefficient \( \frac{1}{2} \) in the function \( y = \frac{1}{2} \tan(x) \). This coefficient affects the amplitude of the graph, reducing the height of the tangent function. Consequently, the points at \( \left(\frac{\pi}{4}, 1\right) \) and \( \left(-\frac{\pi}{4}, -1\right) \) are adjusted to \( \left(\frac{\pi}{4}, \frac{1}{2}\right) \) and \( \left(-\frac{\pi}{4}, -\frac{1}{2}\right) \), respectively. The graph appears shorter, reflecting this change.

Now, we need to incorporate the horizontal shift caused by the term \( -\frac{\pi}{2} \) in the function \( y = \frac{1}{2} \tan(x - \frac{\pi}{2}) \). The shift can be determined using the formula for horizontal transformations, where \( h \) is the shift value and \( b \) is the coefficient of \( x \). Here, \( h = \frac{\pi}{2} \) and \( b = 1 \), leading to a shift of \( \frac{\pi}{2} \) units to the right.

To visualize this shift, we move the asymptotes and key points of the graph. The asymptote at \( x = \frac{\pi}{2} \) shifts to \( x = \pi \), and the asymptote at \( x = -\frac{\pi}{2} \) shifts to \( x = 0 \). This results in new asymptotes at \( x = 0 \) and \( x = \pi \), with an additional asymptote at \( x = -\pi \) due to the periodic nature of the tangent function.

Finally, the graph of \( y = \frac{1}{2} \tan(x - \frac{\pi}{2}) \) will repeat every \( \pi \) units, maintaining the same shape as the original tangent graph but adjusted for the amplitude and horizontal shift. The resulting graph will exhibit the characteristic up-and-down behavior of the tangent function, with the new asymptotes and points reflecting the transformations applied.

Introduction to Cotangent Graph

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 zeros occur at integer multiples of π, specifically at:

• $$ x = n\pi $$ (where n is an integer)

Thus, the cotangent graph will have asymptotes at:

• $$ x = 0, \pm \pi, \pm 2\pi, \ldots $$

Next, the cotangent graph exhibits a repeating pattern similar to the tangent graph, but the curves are inverted. While the tangent graph has curves that rise from the bottom left to the top right, the cotangent graph's curves descend from the top left to the bottom right. The period of both the tangent and cotangent functions is:

$$ \text{Period} = \frac{\pi}{b} $$

For the cotangent function, if the equation is in the form $$ \cot(bx) $$, the period can be calculated using the coefficient b. For example, if b = 1, the period is π.

When sketching the cotangent graph, it is important to note the x-intercepts, which occur at odd multiples of π/2. These points correspond to where the tangent function approaches infinity or negative infinity, leading to the cotangent function equating to zero. The x-intercepts for the cotangent graph are located at:

• $$ x = \frac{(2n-1)\pi}{2} $$ (where n is an integer)

To illustrate, if we consider the cotangent function $$ \cot(\pi x) $$, the period is calculated as:

$$ \text{Period} = \frac{\pi}{\pi} = 1 $$

This means that the asymptotes will occur at every integer value along the x-axis, specifically at x = 0, ±1, ±2, etc. The curves will then be drawn between these asymptotes, descending from the left to the right.

In summary, the cotangent graph is characterized by its vertical asymptotes at integer multiples of π, a period of π/b, and x-intercepts at odd multiples of π/2. By understanding these key features, sketching the cotangent graph becomes a straightforward process.

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

• A. $b=\frac14$
• B. $b=1$
• C. $b=2$
• D. $b=\frac12$

Answer: (A)

Introduction to Cotangent Graph Example 2

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. The term \( \frac{1}{4} \) inside the cotangent function affects the period of the graph.

First, let's focus on the basic cotangent function \( y = \cot\left(\frac{1}{4}x\right) \). The period of a cotangent function is given by the formula:

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

where \( b \) is the coefficient of \( x \). In this case, \( b = \frac{1}{4} \), so the period becomes:

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

The cotangent function has vertical asymptotes at \( x = 0 \), \( x = 4\pi \), and \( x = 8\pi \), repeating every \( 4\pi \) units. The graph of the cotangent function typically decreases from left to right, crossing the x-axis at the midpoint between the asymptotes, which are located at \( \pi \) and \( 3\pi \) for the first period.

Next, we incorporate the vertical stretch caused by the coefficient of 2. This means that instead of reaching \( 1 \) at \( \pi \), the graph will reach \( 2 \), and instead of reaching \( -1 \) at \( 3\pi \), it will reach \( -2 \). Thus, the points on the graph will be adjusted accordingly.

Finally, the negative sign in front of the function flips the graph over the x-axis. Therefore, the point that was at \( ( \pi, 2 ) \) will now be at \( ( \pi, -2 ) \), and the point that was at \( ( 3\pi, -2 ) \) will now be at \( ( 3\pi, 2 ) \).

Putting all these transformations together, 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 \( (2\pi, 0) \), and continue to the upper right, repeating this pattern between each pair of asymptotes. The final graph will reflect these characteristics, demonstrating the periodic nature of the cotangent function while incorporating the effects of amplitude and reflection.

Here’s what students ask on this topic:

What are the key differences between the graphs of secant and cosecant functions?

The key differences between the graphs of secant and cosecant functions lie in the placement of their asymptotes and their relationship to sine and cosine. The secant function is the reciprocal of cosine, so its asymptotes occur where the cosine function equals zero, which are at odd multiples of π/2 (e.g., π/2, 3π/2). On the other hand, the cosecant function is the reciprocal of sine, so its asymptotes occur where the sine function equals zero, which are at integer multiples of π (e.g., 0, π, 2π). Both graphs exhibit repeating "smiley" and "frowny" shapes between their asymptotes, but their asymptote locations are distinct due to their reciprocal relationships with sine and cosine.

How do you determine the period of a tangent or cotangent function?

The period of a tangent or cotangent function is determined using the formula $\frac{\pi }{b}$, where $b$ is the coefficient of $x$ in the function. For example, in $y=\tan \left(\frac{\pi }{2}x\right)$, $b$ is $\frac{\pi }{2}$. The period is calculated as $\frac{\pi }{\frac{\pi }{2}}$, which simplifies to 2. This means the graph repeats every 2 units. For cotangent, the same formula applies, but the graph is flipped compared to tangent.

Why do secant and cosecant graphs have asymptotes?

Secant and cosecant graphs have asymptotes because they are the reciprocals of cosine and sine, respectively. Asymptotes occur at points where the sine or cosine function equals zero, as division by zero is undefined. For the secant function, asymptotes appear at odd multiples of π/2 (e.g., π/2, 3π/2), where cosine equals zero. For the cosecant function, asymptotes occur at integer multiples of π (e.g., 0, π, 2π), where sine equals zero. These asymptotes represent points where the secant or cosecant function approaches infinity or negative infinity, creating vertical lines on the graph.

How do transformations affect the graphs of tangent and cotangent functions?

Transformations affect the graphs of tangent and cotangent functions in similar ways to other trigonometric functions. Vertical stretches or compressions occur when the function is multiplied by a constant. Horizontal stretches or compressions are determined by the coefficient $b$ in the function, which changes the period using the formula $\frac{\pi }{b}$. Phase shifts occur when a constant is added or subtracted inside the function, shifting the graph left or right. Vertical shifts occur when a constant is added or subtracted outside the function, moving the graph up or down. These transformations allow for precise adjustments to the graph's shape and position.

What is the relationship between the tangent and cotangent graphs?

The tangent and cotangent graphs are closely related but have key differences. Both have a period of $\pi$ and repeat every $\pi$ units. However, the tangent graph has asymptotes at odd multiples of $\frac{\pi }{2}$ (e.g., $\frac{\pi }{2}$, $\frac{3}{\pi }$), while the cotangent graph has asymptotes at integer multiples of $\pi$ (e.g., 0, $\pi$, $2\pi$). Additionally, the tangent graph increases from bottom left to top right, while the cotangent graph decreases from top left to bottom right, making them mirror images of each other.

How do you graph the cosecant function using the sine function?

To graph the cosecant function using the sine function, follow these steps: 1) Start by graphing the sine function, as the cosecant is its reciprocal. 2) Identify the points where the sine function equals zero (e.g., 0, π, 2π). These are the locations of the vertical asymptotes for the cosecant graph, as division by zero is undefined. 3) At the peaks and valleys of the sine graph, plot the corresponding points for the cosecant graph, as the reciprocal of 1 is 1 and the reciprocal of -1 is -1. 4) Draw the "smiley" and "frowny" curves of the cosecant graph between the asymptotes, ensuring they approach infinity near the asymptotes. This method leverages the relationship between sine and cosecant for accurate graphing.