Graphing Sine, Cosine, Cosecant, Secant, Tangent & Cotangent (Complete Guide)

Mario's Math Tutoring

19 Aug 201930:40

EducationalLearning

32 Likes 10 Comments

TLDRThis comprehensive video tutorial guides viewers through graphing the fundamental trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent. It begins with basic graphs and progresses to more complex transformations, including amplitude changes, period adjustments, and phase shifts. The video also references the unit circle for graph origins and provides a clear explanation of how to calculate periods and shifts for each function. The tutorial is designed to help learners understand and apply the principles of trigonometry to graphing, with time stamps provided for easy navigation through the content.

Takeaways

📈 The basic sine graph can be graphed by considering the y-coordinate of points on the unit circle at different angles.
📊 The amplitude (a) in the function y = a * sine(X) affects the vertical stretch of the graph, with a > 1 causing an upward stretch and 0 < a < 1 causing a downward compression.
🔄 The period (B) in the function y = sine(B * X) changes the frequency of the sine wave; the period is given by 2π/B.
🔄 The phase shift in the function y = sine(X + C) affects the horizontal positioning of the graph, with a positive C shifting the graph to the left and a negative C to the right.
📉 The basic cosine graph starts at the maximum value, decreases to the midline, reaches the minimum, returns to the midline, and increases back to the maximum, repeating this pattern.
🔄 The period of the cosine function can also be altered by the coefficient B in front of X, with the new period being 2π/B.
🔄 The vertical shift (K) in the function y = cosine(X) + K affects the overall vertical position of the graph, with a positive K shifting the graph up and a negative K down.
🔄 The secant and cosecant functions are reciprocals of the cosine and sine functions, respectively, leading to different asymptotic behaviors in their graphs.
📈 The tangent function has a period of π and its graph includes vertical asymptotes at odd multiples of π/2, with the graph repeating every interval of π.
🔄 The cotangent function is the reciprocal of the tangent and has a period of π, with its graph extending from 0 to π and including vertical asymptotes at even multiples of π/2.

Q & A

What is the basic shape of the sine graph?
-The basic shape of the sine graph starts at the midline, goes up to the maximum (1), back to the midline, down to the minimum (-1), and then back to the midline, where it repeats this pattern.
How does the amplitude affect the graph of a sine function?
-The amplitude is a vertical stretch or compression of the graph. If the amplitude is greater than 1, it stretches the graph vertically, and if it is between 0 and 1, it compresses the graph vertically.
What is the period of a sine function and how is it affected by the value of 'b' in the function y = sine(bx)?
-The period of a sine function is the interval over which the function completes one full cycle. In the function y = sine(bx), the period is affected by the value of 'b' and is calculated as the reciprocal of 'b', i.e., period = 2π / |b|.
What is the basic shape of the cosine graph?
-The basic shape of the cosine graph starts at the maximum (1), goes down to the midline, continues down to the minimum (-1), returns to the midline, and then goes back up to the maximum (1), where it repeats the pattern.
How does the phase shift affect the graph of a sine or cosine function?
-A phase shift affects the horizontal position of the graph. Adding π (or subtracting 2π in the case of cosine) shifts the graph to the left, while subtracting π (or adding 2π for cosine) shifts it to the right. A vertical shift moves the graph up or down, depending on whether the shift value is positive or negative.
What are the asymptotes of the secant and cosecant functions?
-The asymptotes of the secant and cosecant functions occur where the cosine and sine functions, respectively, cross the x-axis. These are the points where the secant and cosecant functions approach but never touch or cross the x-axis.
How does the coefficient 'h' in the function y = A * sine(Bx - h) affect the graph?
-The coefficient 'h' in the function y = A * sine(Bx - h) affects the horizontal shift of the graph. A positive 'h' value shifts the graph to the right, while a negative 'h' value shifts it to the left.
What is the basic shape of the tangent graph?
-The basic shape of the tangent graph consists of alternating vertical asymptotes and lines that approach but never touch the asymptotes, increasing and decreasing in a symmetrical pattern around the y-axis.
How does the coefficient 'k' in the function y = A * tangent(Bx + k) affect the graph?
-The coefficient 'k' in the function y = A * tangent(Bx + k) affects the vertical shift of the graph. A positive 'k' value shifts the graph upwards, while a negative 'k' value shifts it downwards.
What are the unit circle values used for cotangent and how do they affect the graph?
-The unit circle values used for cotangent range from 0 to π (0 to 180 degrees). These values affect the graph by creating vertical asymptotes at points where the cosine value is 0 (π/2 and 3π/2), as cotangent is undefined at these points.

Outlines

00:00

📊 Introduction to Trigonometric Graphs

This paragraph introduces the video's purpose, which is to provide a comprehensive guide on graphing sine, cosine, tangent, secant, cosecant, and cotangent functions. It begins with a discussion on how to graph basic sine and cosine graphs and progresses to more complex topics like transformations (shifting, stretching, compressing) and the role of the unit circle in understanding these graphs. The paragraph emphasizes the importance of understanding the basic shapes of these graphs for solving more challenging problems.

05:01

📈 Graphing Sine and Cosine Functions

The paragraph delves into the specifics of graphing sine and cosine functions, explaining how to plot points on the unit circle and how these points translate into the graphs. It covers the effects of amplitude changes on the vertical stretch of the graphs and the impact of the 'B' value on the period of the functions. The paragraph also introduces the concept of phase shifts and vertical shifts, using examples to illustrate how these transformations affect the graphs.

10:02

🔄 Understanding Period and Phase Shifts

This section focuses on the concepts of period and phase shifts in trigonometric functions. It explains how to calculate the period using the 'B' value and how phase shifts affect the position of the graph on the coordinate axis. The paragraph uses examples of sine and cosine functions with different 'B' values and phase shifts to demonstrate the changes in their periods and the shifts in their starting points.

15:04

📊 Graphing Secant and Cosecant Functions

The paragraph discusses the reciprocal nature of secant and cosecant functions in relation to cosine and sine, respectively. It explains how to graph these functions, including the identification of vertical asymptotes where the cosine or sine function equals zero. The paragraph also covers the impact of coefficients and 'B' values on the shape and period of secant and cosecant graphs, using examples to illustrate these concepts.

20:06

📈 Advanced Trigonometric Graphs

This section tackles more complex trigonometric graphs involving multiple transformations. It explains how to graph functions with combined amplitude, period, phase shift, and vertical shift changes. The paragraph uses specific examples to demonstrate how these transformations affect the graph, including the calculation of new periods and the visualization of phase and vertical shifts on the graphs.

25:06

📊 Graphing Tangent and Cotangent Functions

The paragraph shifts focus to tangent and cotangent functions, explaining their relationship with the unit circle and the specific values used for these functions. It discusses the graphing of tangent and cotangent functions, including the identification of vertical asymptotes and the impact of coefficients on the vertical stretch or shrink of the graphs. The paragraph also covers the effects of 'B' values on the period of these functions and provides examples to illustrate these concepts.

30:07

📈 Complex Trigonometric Graphs and Shifts

This section deals with more complex scenarios in trigonometric graphing, particularly focusing on the combined effects of amplitude, period, and phase shifts on tangent and cotangent graphs. The paragraph provides a detailed explanation of how to calculate the period and how to visualize the shifts on the graphs. It uses specific examples to demonstrate the transformations and their impact on the overall shape and position of the graphs.

🎓 Conclusion and Channel Promotion

The final paragraph wraps up the video by encouraging viewers to subscribe to the channel for more content on making math learning less stressful. It reiterates the channel's goal of helping viewers improve their math grades and achieve their academic dreams. The paragraph serves as a call to action for viewers to engage with the channel for future educational content.

Mindmap

Keywords

💡Sine Function

The sine function is a fundamental mathematical function that describes the periodic behavior of oscillations and vibrations. In the context of the video, it is used to graph the basic sine wave, which starts at the midline, reaches a maximum, returns to the midline, goes to a minimum, and then repeats this pattern. The sine function is represented as y = sin(x), and its graph is a wave that oscillates between -1 and 1, crossing the x-axis at multiples of π.

💡Cosine Function

The cosine function is another fundamental periodic function, closely related to the sine function, used to model periodic phenomena. In the video, the cosine graph is explained as starting at its maximum value, descending to the midline, reaching a minimum, returning to the midline, and then ascending back to the maximum, similar to the sine wave but phase-shifted by π/2 radians or 90 degrees.

💡Tangent Function

The tangent function is a trigonometric function that represents the ratio of the sine to the cosine of an angle. It is defined for all real numbers except where it is undefined due to division by zero. In the video, the tangent graph is characterized by its sharp peaks and valleys, with asymptotic lines where the function approaches but never reaches the x-axis, occurring at odd multiples of π/2.

💡Period

The period of a function is the interval over which the function's graph completes one full cycle before repeating its pattern. In trigonometric functions like sine and cosine, the standard period is 2π. The video explains how the period can be altered by a coefficient (B value) in the function, such as in y = sin(2x), which has a period of π.

💡Amplitude

Amplitude refers to the maximum distance from the midline (or mean position) of a periodic wave to its peak. In the context of the video, amplitude is discussed in relation to how it affects the vertical stretch of the sine and cosine graphs. For instance, y = 2sin(x) indicates a graph that is vertically stretched by a factor of 2 compared to the basic sine graph.

💡Phase Shift

A phase shift in a periodic function is a horizontal shift that changes the point at which the function starts its cycle. In the video, phase shifts are described with expressions like y = sin(x + π), which indicates a graph that has been shifted to the left by π units, or y = sin(x - π), which would shift the graph to the right by π units.

💡Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches but never actually intersects or crosses. In trigonometric functions, vertical asymptotes occur at values of x where the denominator of a fraction (such as in the case of tangent or cotangent functions) becomes zero, leading to an undefined value and a sharp turn in the graph.

💡Secant Function

The secant function is the reciprocal of the cosine function, defined as 1 divided by the cosine of an angle. In the video, the secant graph is described as having vertical asymptotes where the cosine function crosses the x-axis and taking on very high or low values as the cosine approaches zero, effectively inverting the shape of the cosine graph.

💡Cosecant Function

The cosecant function is the reciprocal of the sine function, defined as 1 divided by the sine of an angle. Similar to the secant, the cosecant graph has vertical asymptotes where the sine function crosses the x-axis. The cosecant graph is characterized by its behavior of approaching infinity and negative infinity as the sine value approaches zero.

💡Cotangent Function

The cotangent function is the reciprocal of the tangent function, calculated as the x-coordinate divided by the y-coordinate (or 1 divided by the tangent). The cotangent graph is similar to the tangent but inverted and shifted, with asymptotes at the y-axis and the behavior of the graph being to go downwards to the right, unlike the tangent which goes upwards to the right.

Highlights

The video provides a comprehensive guide on graphing sine, cosine, tangent, secant, cosecant, and cotangent functions.

The basic sine graph is explained with key points at 0, 45, 90, 180, 270, and 360 degrees.

The amplitude affects the vertical stretch of the graph, doubling the amplitude changes the peak values from 1 to 2.

The period of a function is affected by the coefficient 'B', with a formula for new period being 'Period = 2π / B'.

The sine function graph is vertically stretched and horizontally compressed to form the graph of y = 2 * sin(2x).

Shifts in the graph are described by the 'h' and 'k' values, shifting the graph left or right and up or down.

The basic cosine graph is explained with key points at 0, 90, 180, 270, and 360 degrees.

The negative sign in front of the cosine function reflects the graph over the x-axis.

Changing the 'B' value in the cosine function affects the period, demonstrating how to find the new period with the formula.

The secant and cosecant functions are the reciprocals of the cosine and sine functions, respectively.

The secant graph has vertical asymptotes where the cosine graph crosses the x-axis.

The tangent function graph is explained with asymptotes and key points within the range of -π/2 to π/2.

The cotangent function graph is explained with asymptotes and its unique behavior compared to tangent, with key points from 0 to π.

The video concludes with more challenging examples involving all components (A, B, h, and k) and their effects on the graphs.

The amplitude and period changes are demonstrated with an example of y = 3 * sin(1/2 * x - π) - 2.

The video emphasizes the importance of understanding the basic shapes of sinusoidal graphs for solving more complex problems.

Transcripts

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