Graphing Trigonometric Functions

[br][b]Let us investigate the parent trigonometric functions and transformations upon them.[br][br][/b]Familiarise yourself with the graphs of the six basic trigonometric functions. Ensure that the values in the applet are:[br][list][*]a=1[/*][*]b=1[/*][*]c=0[/*][*]d=0[/*][/list]
_________________________________________________________________[br][br][size=150][b][color=#0000ff]The Sine Function: [math]\large\sf{g(x)=\sin(x)}[/math][/color][/b][/size][br][br]In the applet, click the slider so that [math]\large\sf{\sin(x)}[/math] appears.
[b][size=150]_________________________________________________________________[br][color=#0000ff]Characteristics of the sine function [/color][math]\large\sf{g(x)=\sin(x).}[/math][/size][/b][br]Trigonometric functions are [b]periodic functions[/b]. Notice that the distance from a peak to the next peak, what we call the [b]period[/b], is [math]\large\sf{2\pi.}[/math] This is also the distance between trough and the next trough.[br][br]The midline is dashed line drawn in teal. The function oscillates evenly about this line.[br][br]State [br][list][*]the domain (use interval notation), [/*][*]the range (use interval notation), [/*][*]the period, [/*][*]the equation for the midline, [/*][*]the value when [math]\large\sf{x=0}[/math] for the sine function, and [/*][*]the function's behaviour (whether [math]\large\sf{g(x)=sin(x)}[/math] is an even or an odd function)[/*][/list][i][size=85]Recall that even functions are symmetric about the y-axis and odd function are symmetric about the origin.[/size][/i][br]
_________________________________________________________________[br][br][size=150][b][color=#0000ff]The Cosine Function: [math]\large\sf{g(x)=\cos(x)}[/math][/color][/b][/size][br][br]In the applet, click the slider one over so that [math]\large\sf{\cos(x)}[/math] appears.
[b][size=150]_________________________________________________________________[br][color=#0000ff][b][size=150][color=#0000ff]Characteristics of the[/color][/size][/b] cosine function [/color][math]\large\sf{g(x)=\cos(x).}[/math][/size][/b][br][br]Notice the similarities and differences between the cosine and sine functions.[br][br]State [br][list][*]the domain (use interval notation), [/*][*]the range (use interval notation), [/*][*]the period, [/*][*]the equation for the midline, [/*][*]the value when [math]\large\sf{x=0}[/math] for the sine function, and [/*][*]the function's behaviour (whether [math]\large\sf{g(x)=cos(x)}[/math] is an even or an odd function)[/*][/list][i][size=85]Recall that even functions are symmetric about the y-axis and odd function are symmetric about the origin.[/size][/i][br]
_________________________________________________________________[br][br][size=150][b][color=#0000ff]The Tangent Function: [math]\large\sf{g(x)=\tan(x)}[/math][/color][/b][/size][br][br]In the applet, click the slider one over so that [math]\large\sf{\tan(x)}[/math] appears.
[b][size=150]_________________________________________________________________[br][color=#0000ff][b][size=150][color=#0000ff]Characteristics of the[/color][/size][/b] tangent function [/color][math]\large\sf{g(x)=\tan(x).}[/math][/size][/b][br][br]Tangent is a composite of the sine and cosine functions: [math]\large\sf{\tan(x)=\frac{\sin(x)}{\cos(x)}}[/math][br][br]Notice the periodic asymptotic behaviour which corresponds to when values of [math]\large\sf{\cos(x)=0.}[/math][br][br]State [br][list][*]the domain (use interval notation), [/*][*]the range (use interval notation), [/*][*]the period, [/*][*]the equation for the midline, [/*][*]the value when [math]\large\sf{x=0}[/math] for the tangent function, and [/*][*]the function's behaviour (whether [math]\large\sf{g(x)=tan(x)}[/math] is an even or an odd function)[/*][/list]
_________________________________________________________________[br][br][size=150][b][color=#0000ff]The Cosecant Function: [math]\large\sf{g(x)=\csc(x)}[/math][/color][/b][/size][br][br]In the applet, click the slider one over so that [math]\large\sf{\csc(x)}[/math] appears.
[b][size=150]_________________________________________________________________[br][color=#0000ff][b][size=150][color=#0000ff]Characteristics of the[/color][/size][/b] cosecant function [/color][math]\large\sf{g(x)=\csc(x).}[/math][/size][/b][br][br]Cosecant is the reciprocal of the sine function: [math]\large\sf{\csc(x)=\frac{1}{\sin(x)}}[/math][br][br]Notice the periodic asymptotic behaviour which corresponds to when values of [math]\large\sf{\sin(x)=0.}[/math][br][br]State [br][list][*]the domain (use interval notation), [/*][*]the range (use interval notation), [/*][*]the period, [/*][*]the equation for the midline, [/*][*]the value when [math]\large\sf{x=0}[/math] for the cosecant function, and [/*][*]the function's behaviour (whether [math]\large\sf{g(x)=csc(x)}[/math] is an even or an odd function)[/*][/list]
_________________________________________________________________[br][br][size=150][b][color=#0000ff]The Secant Function: [math]\large\sf{g(x)=\sec(x)}[/math][/color][/b][/size][br][br]In the applet, click the slider one over so that [math]\large\sf{\sec(x)}[/math] appears.
[b][size=150]_________________________________________________________________[br][color=#0000ff][b][size=150][color=#0000ff]Characteristics of the[/color][/size][/b] secant function [/color][math]\large\sf{g(x)=\sec(x).}[/math][/size][/b][br][br]Secant is the reciprocal of the cosine function: [math]\large\sf{\sec(x)=\frac{1}{\cos(x)}}[/math].[br][br]Notice the periodic asymptotic behaviour which corresponds to when values of [math]\large\sf{\cos(x)=0.}[/math][br][br][br]State [br][list][*]the domain (use interval notation), [/*][*]the range (use interval notation), [/*][*]the period, [/*][*]the equation for the midline, [/*][*]the value when [math]\large\sf{x=0}[/math] for the secant function, and [/*][*]the function's behaviour (whether [math]\large\sf{g(x)=sec(x)}[/math] is an even or an odd function)[/*][/list]
_________________________________________________________________[br][br][size=150][b][color=#0000ff]The Cotangent Function: [math]\large\sf{g(x)=\cot(x)}[/math][/color][/b][/size][br][br]In the applet, click the slider one over so that [math]\large\sf{\cot(x)}[/math] appears.
[b][size=150]_________________________________________________________________[br][color=#0000ff][b][size=150][color=#0000ff]Characteristics of the[/color][/size][/b] cotangent function [/color][math]\large\sf{g(x)=\cot(x).}[/math][/size][/b][br][br]Cotangent is the reciprocal of the tangent function, and hence it is also a composite of the sine and cosine functions: [math]\large\sf{\cot(x)=\frac{1}{\tan(x)}=\frac{\cos(x)}{\sin(x)}}[/math][br][br]Notice the periodic asymptotic behaviour which corresponds to when values of [math]\large\sf{\sin(x)=0.}[/math][br][br]State [br][list][*]the domain (use interval notation), [/*][*]the range (use interval notation), [/*][*]the period, [/*][*]the equation for the midline, [/*][*]the value when [math]\large\sf{x=0}[/math] for the cotangent function, and [/*][*]the function's behaviour (whether [math]\large\sf{g(x)=cot(x)}[/math] is an even or an odd function)[/*][/list]
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– Stephanie Kong

Information: Graphing Trigonometric Functions