L4 Trigonometric Function Graphing Review - Part 3
Completion requirements
Unit 4A
Trigonometry Part 1
Lesson 4: Trigonometric Function Graphing Review
The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math»-values used to graph the functions on the previous page were restricted between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math». However, it is possible to include angles outside
this domain. The graphs of the sine and cosine functions continue indefinitely in both directions. Note both graphs have the same shape. Both functions are referred to as sinusoidal.
Since any real angle measure can be used as an input for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», the domain for each function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»|«/mi»«mo»§#952;«/mo»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The graph’s pattern repeats regularly, and is said to be periodic. The length of one cycle in the graph’s pattern is called the function’s period. The period for the sine and cosine function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math».
The range for each function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»f«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«mi mathvariant=¨normal¨»|«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#8804;«/mo»«mo»§#952;«/mo»«mo»§#8804;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#952;«/mo»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math» since the radius of the unit circle cannot be greater than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» or less than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math».
The range is closely related to amplitude, which is half the distance from the minimum to the maximum, or the distance between the midline and either the maximum or minimum.
Since any real angle measure can be used as an input for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», the domain for each function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»|«/mi»«mo»§#952;«/mo»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The graph’s pattern repeats regularly, and is said to be periodic. The length of one cycle in the graph’s pattern is called the function’s period. The period for the sine and cosine function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math».
The range for each function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»f«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«mi mathvariant=¨normal¨»|«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#8804;«/mo»«mo»§#952;«/mo»«mo»§#8804;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#952;«/mo»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math» since the radius of the unit circle cannot be greater than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» or less than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math».
The range is closely related to amplitude, which is half the distance from the minimum to the maximum, or the distance between the midline and either the maximum or minimum.