Trigonometric Functions (Graphing & Modeling)
Trigonometric Functions Applets
Table of Contents
- Prerequisite Topics
- 1 Radian: Clear Definition
- Movie: Radians to Revs
- Trigonometric Functions of Special Angles
- Angles in Standard Position
- Angle Orientations (Standard Position)
- Coterminal Angles Action!!!
- Coterminal Angles: Question Generator
- Trig Function Values (30 Degrees)
- Trig Function Values (45 Degrees)
- Trig Function Values (60 Degrees)
- Special Angles in Standard Position: Intro (V2)
- Special Angles in Standard Position: Intro (V1)
- Special Angles in Standard Position: Intro (V3)
- Quiz 1: Trigonometric Functions of Special Angles
- Quiz 2: Trigonometric Functions of Special Angles
- Graphing Trigonometric Functions
- Periodic Function Action!
- Sine & Cosine Period Action (1)!
- Sine & Cosine Period Action (2)!
- Tangent Period Action (2)!
- Unit Circle to Sine and Cosine Functions
- Tangent Function Exploration
- Cotangent Function Exploration
- Graphing Sine & Cosine Functions (II)
- Quiz: Graphing Sine & Cosine Functions (Amplitude & Vertical Shift Only)
- Graphing Sine & Cosine Functions (I)
- Tangent Period Action (1)!
- Inverse Trigonometric Functions
- Sine Function Domain Restriction Options?
- Cosine Function Domain Restriction Options?
- Tangent Function Domain Restriction Options?
- Modeling with Trigonometric Functions
- YOUR Linear Speed?
- How Fast are You Spinning?
- Writing Sine and Cosine Functions (with Amplitude & Vertical Shift Changes)
- Sine Function Modeling (y = a sin(x) + d) Quiz
- Cosine Function Modeling (y = a cos(x) + d) Quiz
- Modeling with Trigonometric Functions
- Sine & Cosine Function Modeling: Finding Correct Parameters
- Ferris Wheel Action!!!
- Ferris Wheel (1): Modeling with Trigonometric Functions
- Ferris Wheel (2): Modeling with Trigonometric Functions
1 Radian: Clear Definition
[color=#c51414]One unit of ANGLE or ARC MEASURE which you're probably familiar with is that of a "degree." One degree is 1/360th of a full revolution, right? [/color][br][color=#0a971e]Another unit of ANGLE or ARC MEASURE is a "revolution". 1 revolution = 360 degrees, right? [/color][br][br][color=#1551b5]Well, there is ANOTHER unit of ANGLE or ARC MEASURE with which you'll soon become familiar. [/color] [br][color=#1551b5]This new unit of ANGLE or ARC MEASURE is called a [b]RADIAN[/b]. [/color] [br][br][i][color=#b20ea8]Interact with the applet below for a few minutes. [br]Reset it a few times and start the animation again each time.[br]Be sure to change the circle's radius as you go along. [br][br][b][color=#1551b5]After interacting with this applet, answer the question that appears immediately below it.[/color][/b][/color] [/i]
Again, recall that a "degree", a "revolution", and a "radian" are all units of ARC MEASURE (i.e. AMOUNT OF SPIN). [br][br][color=#c51414][b]Complete the following sentence definition:[/b][/color] [br][br][b][color=#1551b5]Definition: 1 RADIAN is defined to be a unit of ARC MEASURE for which.....[/color][/b]
Angles in Standard Position
The angle drawn below in the coordinate plane is classified as being drawn in [b]STANDARD POSITION. [br][br][/b]Interact with the applet for a minute.[br]Then answer the question that follows.
ANGLE IN STANDARD POSITION:
1.
What does it mean for an angle drawn in the coordinate plane to be drawn in [b]STANDARD POSITION? [br][br][/b](Your definition should list 2 criteria.)
Periodic Function Action!
The applet below dynamically depicts what it means for a function to be classified as a [color=#9900ff][b]periodic[/b][/color][b] function. [br][br][/b]Interact with the applet below for a few minutes. [br]As you do, be sure to move the 5 points anywhere you'd like. [br]Answer the questions that follow.
1.
What does the term "periodic" mean? What events in life would you classify as predictably "periodic"? Why would you classify these events this way?
2.
The [b][color=#9900ff]length of the purple segment[/color][/b] is said to be the [b][color=#9900ff]period[/color][/b] of the function [i]f[/i]. Suppose the [b][color=#9900ff]period[/color] [/b]of the function above = [b][color=#9900ff]5 units[/color][/b]. Also, suppose [i]f[/i](2) = 8. Given this information, [b]what other input values for function [i]f[/i] would also result in an output of 8?[/b]
3.
Suppose a function [i]f[/i] has [b][color=#9900ff]period [i]c[/i][/color][/b]. How would you describe what it means for a function [i]f[/i] to be periodic in terms of [i]x[/i] and [b][color=#9900ff]c[/color][/b]?
Sine Function Domain Restriction Options?
Recall that in order for a relation to be a function, [b]each & every input can have one and only one output[/b]. [br][br]The [b]sine function, part of whose graph is displayed below,[/b] is a function because each input (angle) can only have one sine ratio to which it maps (see [url=https://www.geogebra.org/m/S2gMrkbD]this applet[/url] by [url=https://www.geogebra.org/orchiming]Anthony C.M. Or[/url]). [br][br]Interact with the app for a few minutes, then answer the questions that follow.
[b]Directions:[/b][br][br]Click the refresh (recycle) icon at the top of the applet. Then select the [b]Show Inverse Relation [/b]checkbox. Drag the [b]x-final slider[/b] slowly to the right and note the [b]inverse relation being graphed[/b] simultaneously as the [b]sine function is being graphed.[/b]
Why is the relation [math]y=sin\left(x\right)[/math] considered to be a function?
Select the [b]Default to Natural Domain of f [/b]checkbox. Then select [b]Show Inverse Relation. [/b]Is this [b]inverse relation [/b](whose equation is [math]x=sin\left(y\right)[/math]) also a function? Explain why or why not.
How could we restrict the domain of the original function [math]f\left(x\right)=sin\left(x\right)[/math] in order for its inverse relation to be a function as well? There are lots of possibilities. Can you find one? Feel free to experiment by inputting values into the Xmin and Xmax input boxes or by using the sliders.)
YOUR Linear Speed?
[b]Students:[/b][br][br]Use this applet to help you complete the [i][color=#0000ff]How Fast Are You Spinning?[/color] [/i]investigation given to you at the beginning of class. [br][br]Some key questions to consider are listed below the applet.
1.
According to NASA's website, what is the Earth's mean equatorial radius?
2.
According to your answer for (1), what would Earth's circumference be?
3.
What would the linear speed of a person who lives on the Equator be? (Round your answer to the nearest 10 miles per hour).
4.
What is YOUR current latitude? (For Southern Hemisphere users, use a negative number. For Northern Hemisphere users, use a positive number.) .
5.
Use your answer for (3) and your result for (4) to determine YOUR LINEAR SPEED in miles/hr. (Round to the nearest 10 miles per hour). For a hint, interact with the applet above.