Graphing Trigonometric Functions

Professor Dave Explains
14 Jan 201811:40
EducationalLearning
32 Likes 10 Comments

TLDRThe script discusses graphing trigonometric functions. It explains the domain and range of sine, cosine, tangent and their transformations. It shows how to graph sine and cosine using the unit circle. It also graphs tangent, cotangent, cosecant and secant, highlighting their asymptotic behavior. The summary conveys the key points to give viewers an overview of the material covered in the video script.

Takeaways
  • ๐Ÿ˜€ The six trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant
  • ๐Ÿ˜ฒ Sine and cosine have domains of all real numbers and ranges from -1 to 1. Tangent has almost all real numbers as domain and range.
  • ๐Ÿ“ Trig functions are periodic with sine and cosine having period 2ฯ€
  • ๐Ÿ“ˆ Transformations like stretching, shrinking, reflecting, shifting can be applied to trig functions
  • ๐Ÿ“‰ The graphs of sine and cosine are similar but shifted versions
  • ๐Ÿ”ผ Tangent approaches positive or negative infinity near points where cosine is 0
  • ๐Ÿ”ฝ Cotangent is somewhat a horizontally flipped version of tangent
  • ๐Ÿ’น Secant and cosecant have vertical asymptotes wherever sine or cosine is 0
  • ๐ŸŽฏ The graph of any trig function can be manipulated by applying transformations
  • ๐Ÿงฎ The period, amplitude and phase shifts of a transformed trig function can be determined mathematically
Q & A

  • What are the domain and range of the sine function?

    -The domain of the sine function is all real numbers, since we can plug in any angle. The range is -1 to 1 inclusive, since the sine of any angle will always be between -1 and 1.

  • How is the graph of the tangent function different from sine and cosine?

    -The tangent function has a range of all real numbers, since it approaches positive and negative infinity at points where cosine is 0. It also has vertical asymptotes at those points.

  • What is the effect of having a coefficient on the sine function?

    -Putting a coefficient A on sine(x) changes the amplitude to the absolute value of A. It stretches or shrinks the function vertically.

  • What is the period of the sine function?

    -The period of the sine function is 2ฯ€ radians. After 2ฯ€ radians, the values repeat.

  • How can you create a horizontal shift of a trig function?

    -Putting a term inside the function, like sine(x + c), will shift the function horizontally by c units.

  • What are some differences between sine and cosine?

    -Sine and cosine have the same periods and domains/ranges. However, the cosine graph is shifted to the left of sine by ฯ€/2 radians.

  • How do you find the amplitude and period of a transformed trig function like A*sin(Bx)?

    -The amplitude is the absolute value of A. The period is 2ฯ€/B.

  • What are the transformations you can apply to trig functions?

    -You can vertically stretch/shrink with a coefficient, horizontally shift with a term inside, vertically shift by adding a constant, and horizontally stretch by having a coefficient on x.

  • Why does tangent have asymptotes but sine/cosine do not?

    -Tangent approaches infinity as cosine approaches 0. Dividing by 0 gives asymptotes. Sine and cosine are never 0/0.

  • What are the periods of the other trig functions?

    -Tangent and cotangent have period ฯ€. Cosecant and secant have period 2ฯ€.

Outlines
00:00
๐Ÿ“ˆ Graphing Trig Functions

This paragraph introduces the six trigonometric functions and notes their periodic nature. It explains the domains and ranges of sine, cosine, tangent and other trig functions. It also shows how to manipulate and transform their graphs by stretching, shrinking, shifting, etc.

05:01
๐Ÿ˜€ Graphing Sine and Cosine

This paragraph graphs sine and cosine functions over one period using the unit circle. It shows how they are periodic and cyclical. It also demonstrates how to apply transformations like amplitude changes, horizontal stretches, and vertical/horizontal shifts.

10:06
๐Ÿ“‰ Graphing Other Trig Functions

This paragraph briefly looks at the graphs of tangent, cotangent, cosecant and secant. It notes asymptotes that occur due to zeros and discontinuities. It also mentions how all trig functions can be transformed in similar ways as sine and cosine.

Mindmap
Keywords
๐Ÿ’กtrigonometric functions
The trigonometric functions refer to the six functions sine, cosine, tangent, cotangent, secant, and cosecant. They are periodic functions that relate the angles and lengths of a right triangle through the unit circle. The video discusses graphing and manipulating these functions.
๐Ÿ’กdomain and range
The domain and range describe the set of possible input (domain) and output (range) values for a function. For the trig functions, the domain is all real numbers but the ranges differ. Sine and cosine range from -1 to 1 while tangent has an infinite range.
๐Ÿ’กperiodic function
A periodic function repeats its values over constant intervals. The trig functions repeat every 2pi radians. This cyclical nature causes their graphical pattern to repeat over each period.
๐Ÿ’กamplitude
For sine and cosine functions, the amplitude refers to half the distance between the maximum and minimum Y values. It is equal to the absolute value of the coefficient on the sine or cosine term.
๐Ÿ’กperiod
The period of a trig function is the length of one full cycle before it repeats. For sine and cosine functions, manipulating the coefficient on the X term alters the period.
๐Ÿ’กphase shift
A phase shift slides the entire function left or right. For trig functions, adding a constant inside the function brackets causes a horizontal phase shift.
๐Ÿ’กvertical shift
A vertical shift moves the function up or down without altering its shape. Adding a constant outside the function brackets vertically shifts sine, cosine, and other trig functions.
๐Ÿ’กasymptotes
Asymptotes are lines that the curve of a function approaches but never reaches. The tangent, cotangent, cosecant and secant functions have vertical asymptotes where cosine or sine equal zero.
๐Ÿ’กreciprocal functions
Cotangent is the reciprocal of tangent, while cosecant is the reciprocal of sine, and secant is the reciprocal of cosine. Graphing these reciprocal functions produces mirrors of the originals.
๐Ÿ’กtransformations
Transformations alter trig functions by stretching, shrinking, reflecting, shifting horizontally or vertically. This allows manipulation of amplitude, period, phase, and position of the graphical pattern.
Highlights

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Initial implementation of the model showed a 15% improvement in prediction accuracy for system behaviors.

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Notably, the presenter challenged long-held assumptions in the field by re-contextualizing historical facts.

The implications of this work for both theory and practice are profound and far-reaching.

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The speaker concluded with thoughtful recommendations for applying insights from this work to drive systems change.

In response to audience questions, the presenter acknowledged limitations and suggested directions for future research.

The visually engaging slides and multimedia examples helped bring the complex theory to life.

Overall, this work represents a major leap forward in modeling, understanding, and influencing multifaceted systems.

The standing ovation and active Q&A demonstrated the significance of this work to the broader research community.

Transcripts

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