Graphs of Trigonometric Functions: Videos & Practice Problems

Understanding the graphs of trigonometric functions like sine, cosine, tangent, secant, cosecant, and cotangent is essential. Sine and cosine exhibit wave patterns, while tangent and cotangent have asymptotes at odd multiples of π/2. The period for tangent and cotangent is π, contrasting with the 2π period of sine and cosine. Transformations such as vertical shifts apply to all functions. Key equations include the reciprocal identities for cosecant and secant, and the relationship of tangent as sine over cosine. Mastering these concepts enhances comprehension of trigonometric behavior and graphing techniques.

Graph of Sine and Cosine Function

Understanding the sine and cosine functions is essential for graphing these trigonometric functions on a standard x-y graph. The sine function corresponds to the y-coordinates on the unit circle, while the cosine function relates to the x-coordinates. As you traverse the unit circle, the values of sine and cosine repeat, leading to a wave-like pattern in their graphs.

To graph the sine function, y = \sin(x), you can start by plotting key points based on the unit circle. For instance, at x = 0, \sin(0) = 0; at x = \frac{\pi}{2}, \sin\left(\frac{\pi}{2}\right) = 1; at x = \pi, \sin(\pi) = 0; and at x = \frac{3\pi}{2}, \sin\left(\frac{3\pi}{2}\right) = -1. Continuing this process reveals that the sine graph oscillates between -1 and 1, creating a smooth wave pattern.

Similarly, the cosine function, y = \cos(x), can be graphed using its corresponding values. At x = 0, \cos(0) = 1; at x = \frac{\pi}{2}, \cos\left(\frac{\pi}{2}\right) = 0; at x = \pi, \cos(\pi) = -1; and at x = \frac{3\pi}{2}, \cos\left(\frac{3\pi}{2}\right) = 0. The cosine graph also exhibits a wave pattern, but it starts at a maximum value of 1, contrasting with the sine graph that begins at 0.

In wave terminology, the highest points of the graph are referred to as crests or peaks, while the lowest points are known as troughs or valleys. These peaks and valleys can vary in height depending on transformations applied to the sine and cosine functions.

One common transformation is the vertical shift, which occurs when a constant k is added to the function. For example, in the function y = \sin(x) + 1, the entire sine graph is shifted upward by 1 unit. This means that the original peaks and valleys will now be at new heights: the peak that was at 1 becomes 2, and the valley that was at -1 becomes 0. Thus, the graph of y = \sin(x) + 1 will reflect these changes, maintaining the wave pattern but elevated by one unit.

By mastering these concepts, you can effectively graph sine and cosine functions and understand how transformations affect their shapes and positions on the graph.

Sketch the function $y=\cos\left(x\right)-1$ on the graph below.

• A.
• B.
• C.
• D.

Answer: (B)

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.

• A. $y=50$
• B. $y=51$
• C. $y=49$
• D. $y=50+\frac{\sqrt3}{2}$

Answer: (C)

Graph of Sine and Cosine Function Example 1

To graph the function \( y = \sin(x) + 3 \), it is beneficial to start with the basic sine function, \( y = \sin(x) \), which is characterized by its wave-like pattern. The sine function oscillates between -1 and 1, with key points occurring at intervals of \( \frac{\pi}{2} \). Specifically, it reaches its maximum value of 1 at \( x = \frac{\pi}{2} \), crosses the x-axis at \( x = \pi \), and reaches its minimum value of -1 at \( x = \frac{3\pi}{2} \).

When we introduce the transformation \( +3 \) in the function \( y = \sin(x) + 3 \), this indicates a vertical shift of the entire sine graph upwards by 3 units. Consequently, the midline of the sine wave, which originally was at \( y = 0 \), is now elevated to \( y = 3 \). The maximum point, previously at \( y = 1 \), shifts to \( y = 4 \), and the minimum point, originally at \( y = -1 \), moves to \( y = 2 \).

To visualize this, we can identify the new coordinates of key points on the graph. The starting point at the center, which is \( (0, 0) \), shifts to \( (0, 3) \). The peak at \( \left(\frac{\pi}{2}, 1\right) \) becomes \( \left(\frac{\pi}{2}, 4\right) \), while the zero crossing at \( (\pi, 0) \) moves to \( (\pi, 3) \). Similarly, the valley at \( \left(\frac{3\pi}{2}, -1\right) \) shifts to \( \left(\frac{3\pi}{2}, 2\right) \).

Continuing this pattern, the graph will maintain its wave-like behavior, oscillating between the new maximum of 4 and the new minimum of 2. The overall shape remains consistent with the sine function, but every point is elevated by 3 units. Thus, the graph of \( y = \sin(x) + 3 \) will exhibit the same periodicity and amplitude as the original sine function, but centered around the line \( y = 3 \).

In summary, the transformation of the sine function through vertical shifting is a fundamental concept in graphing trigonometric functions, allowing for the adjustment of the graph's position without altering its inherent wave characteristics.

Graphs of Secant and Cosecant Functions

In trigonometry, the cosecant and secant functions are essential extensions of the sine and cosine functions, respectively. Understanding these functions begins with recognizing that the cosecant is the reciprocal of the sine function, expressed mathematically as:

\[ \text{cosecant}(x) = \frac{1}{\sin(x)} \]

Similarly, the secant function is defined as:

\[ \text{secant}(x) = \frac{1}{\cos(x)} \]

When graphing these functions, it is crucial to identify where the sine and cosine functions equal zero, as these points will correspond to vertical asymptotes in the cosecant and secant graphs. Specifically, the cosecant function is undefined wherever the sine function is zero, leading to asymptotes at:

\[ x = n\pi \quad (n \in \mathbb{Z}) \]

For the secant function, the asymptotes occur where the cosine function is zero, resulting in asymptotes at:

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

Graphically, the cosecant function will exhibit a pattern of "smiley" and "frowny" faces between these asymptotes, reflecting the behavior of the sine function. As the sine values approach zero, the cosecant values approach infinity, creating the characteristic vertical asymptotes. Conversely, the secant function mirrors this behavior around its own asymptotes, which are located at odd multiples of \(\frac{\pi}{2}\).

To graph a transformed cosecant function, such as \(y = \text{cosecant}(2x)\), one should first graph the corresponding sine function, \(y = \sin(2x)\). The period of the sine function can be determined using the formula:

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

where \(b\) is the coefficient of \(x\). In this case, \(b = 2\), leading to a period of \(\pi\). The sine graph will complete one full cycle from \(0\) to \(\pi\), with peaks and troughs occurring at regular intervals. The cosecant graph will then be constructed by identifying the peaks of the sine graph, which correspond to the defined values of the cosecant, and placing vertical asymptotes at the zeros of the sine function.

In summary, the cosecant and secant functions can be effectively graphed by leveraging the properties of their respective sine and cosine functions, utilizing asymptotes to indicate undefined values, and applying transformation rules for shifts and stretches. This approach allows for a comprehensive understanding of their behavior and graphical representation.

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 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 (0), we find that:

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

At \( x = \frac{\pi}{4} \), the tangent function yields:

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

Conversely, at \( x = -\frac{\pi}{4} \), we have:

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

However, 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 tangent graph exhibits periodic behavior, repeating every \( \pi \) units, unlike the secant function, which has a period of \( 2\pi \). The asymptotes for the tangent function occur at odd multiples of \( \frac{\pi}{2} \), specifically at:

\[ x = \frac{(2n+1)\pi}{2} \quad \text{for } n \in \mathbb{Z} \]

To determine the period of a transformed tangent function, such as \( y = \tan\left(\frac{\pi}{2}x\right) \), we use the formula for the period of tangent:

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

Here, \( b = \frac{\pi}{2} \), leading to:

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

This indicates that the graph will repeat every 2 units along the x-axis. Asymptotes can be placed at \( x = -1 \) and \( x = 1 \), with additional asymptotes at \( x = 3 \) and \( x = -3 \) due to the periodic nature of the function. The resulting graph will curve upwards to the right and downwards to the left, creating a characteristic shape between each pair of asymptotes.

Understanding these properties of the tangent function allows for effective graphing and analysis, reinforcing the connections between trigonometric functions and their graphical representations.

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 modify the function to \( y = \frac{1}{2} \tan(x) \). The coefficient \( \frac{1}{2} \) 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. This results in a graph that is shorter than the standard tangent graph.

Now, we 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 \( \frac{h}{b} \), where \( h \) is the horizontal shift 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 take the asymptotes and points from the previous graph and move them \( \frac{\pi}{2} \) units to the right. 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 \( x = -\pi \).

Finally, the graph will continue to repeat every \( \pi \) units, maintaining the same shape as the original tangent function 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 expressed mathematically as:

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

To sketch the cotangent graph, it is crucial to identify the locations of the asymptotes. The tangent function has zeros at odd multiples of \(\frac{\pi}{2}\), specifically at points such as \(-\frac{3\pi}{2}\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), and \(\frac{3\pi}{2}\). Since the cotangent function cannot take these values (as division by zero is undefined), the asymptotes for the cotangent graph will be located at:

\[ x = n\pi \quad (n \in \mathbb{Z}) \]

This means there will be asymptotes at \(x = -\pi\), \(0\), and \(\pi\), with additional asymptotes at integer multiples of \(\pi\) as the graph extends. The cotangent graph exhibits a repeating pattern every \(\pi\) units, similar to the tangent graph, which also has a period of \(\pi\). The period can be calculated using the formula:

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

In this case, if \(b = 1\), the period simplifies to \(1\). This means that the cotangent graph will repeat every unit interval 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 \(0\) and \(1\), the curve will start high on the left and drop down to the right. This behavior continues for each segment defined by the asymptotes.

In summary, the cotangent graph is characterized by its asymptotes at integer multiples of \(\pi\), a period of \(\pi\), and curves that descend from left to right. Recognizing these features allows for a straightforward sketching process, reinforcing the relationship between cotangent and tangent functions.

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 1

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.

First, let's focus on the basic cotangent function \( y = \cot\left(\frac{1}{4}x\right) \). The period of the cotangent function is calculated using 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 graph will repeat every \( 4\pi \) units. The cotangent function has vertical asymptotes at \( x = 0 \), \( x = 4\pi \), and \( x = 8\pi \). The function approaches infinity as it nears these asymptotes, and it crosses the x-axis at the midpoint between them, specifically at \( x = 2\pi \).

Next, we identify key points on the graph. The cotangent function typically has outputs of 1 and -1 at \( x = \pi \) and \( x = 3\pi \), respectively. Therefore, for the function \( y = \cot\left(\frac{1}{4}x\right) \), we have:

\[ y(\pi) = 1 \quad \text{and} \quad y(3\pi) = -1 \]

Now, incorporating the vertical stretch by a factor of 2, the outputs change to:

\[ y(\pi) = 2 \quad \text{and} \quad y(3\pi) = -2 \]

Finally, we apply the reflection over the x-axis due to the negative sign in front of the cotangent function. This reverses the y-values, resulting in:

\[ y(\pi) = -2 \quad \text{and} \quad y(3\pi) = 2 \]

Thus, 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, maintaining the periodic nature of the cotangent function. The complete graph will exhibit this behavior between each pair of asymptotes, creating a repeating pattern.

Here’s what students ask on this topic:

How do you graph the sine and cosine functions?

To graph the sine and cosine functions, start by plotting key points using the unit circle values. For the sine function, y = sin(x), plot points at (0,0), (π/2,1), (π,0), (3π/2,-1), and (2π,0). Connect these points with a smooth, wave-like curve. For the cosine function, y = cos(x), plot points at (0,1), (π/2,0), (π,-1), (3π/2,0), and (2π,1). Again, connect these points with a smooth curve. Both functions exhibit a repeating wave pattern with a period of 2π. The sine function starts at (0,0), while the cosine function starts at (0,1).

What are the transformations that can be applied to sine and cosine graphs?

Transformations that can be applied to sine and cosine graphs include vertical shifts, horizontal shifts, amplitude changes, and period changes. A vertical shift is achieved by adding a constant k to the function, y = sin(x) + k or y = cos(x) + k, shifting the graph up if k is positive and down if k is negative. Horizontal shifts involve adding or subtracting a constant inside the function, y = sin(x - h) or y = cos(x - h), shifting the graph right if h is positive and left if h is negative. Amplitude changes are made by multiplying the function by a constant a, y = a*sin(x) or y = a*cos(x), affecting the height of the wave. Period changes are made by multiplying the variable x by a constant b, y = sin(bx) or y = cos(bx), altering the length of one complete cycle.

How do you graph the secant and cosecant functions?

To graph the secant and cosecant functions, start by graphing their reciprocal functions, cosine and sine, respectively. For y = sec(x), graph y = cos(x) first, then draw vertical asymptotes at the x-values where cos(x) = 0. The secant graph will have U-shaped curves opening upwards or downwards between these asymptotes. For y = csc(x), graph y = sin(x) first, then draw vertical asymptotes at the x-values where sin(x) = 0. The cosecant graph will have U-shaped curves opening upwards or downwards between these asymptotes. Both functions exhibit periodic behavior with the same period as their reciprocal functions, 2π.

What is the period of the tangent and cotangent functions?

The period of the tangent and cotangent functions is π. For the tangent function, y = tan(x), the graph repeats every π units, with vertical asymptotes at odd multiples of π/2. For the cotangent function, y = cot(x), the graph also repeats every π units, with vertical asymptotes at integer multiples of π. This periodicity is different from the sine and cosine functions, which have a period of 2π. The shorter period of π for tangent and cotangent results in more frequent repeating patterns on the graph.

How do you graph the tangent function?

To graph the tangent function, y = tan(x), start by plotting key points using the identity tan(x) = sin(x)/cos(x). Plot points at (0,0), (π/4,1), and (-π/4,-1). Draw vertical asymptotes at x = ±π/2, where the function is undefined. The graph will have a repeating pattern with a period of π, curving upwards to the right and downwards to the left between each pair of asymptotes. The tangent function has no maximum or minimum values, and the graph extends infinitely in both directions.