L2.3 A3 Graphing the Cosine Function
Completion requirements
Unit 2
Trigonometry
Graphing the Cosine Function
The cosine function, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«mo»=«/mo»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», is very similar to the sine function, and can be thought of as using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values from the unit circle, instead of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values. This can be visualized by rotating the unit circle such that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» occurs at the bottom of the circle and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» occurs at the top.This diagram shows «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»P«/mi»«mfenced»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» corresponds to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
Aside from the starting location, the graphs of the sine function and the cosine function have the same shape. Both functions are referred to as sinusoidal.
Like other functions, any variable can be used to represent the input of a sinusoidal function, and you may see a function represented as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math». Be careful when interpreting a function written this way. On the unit circle, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»y«/mi»«/mrow»«/mfenced»«/mstyle»«/math» represents the intersection between the unit circle and the terminal arm of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math». In «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» represents an angle and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» represents the value of the corresponding ratio.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable»«mtr»«mtd»«mtable columnalign=¨right center right center center center¨»«mtr»«mtd»«mi mathcolor=¨#FF0000¨»point«/mi»«mo mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathcolor=¨#FF0000¨»of«/mi»«mo mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathcolor=¨#FF0000¨»intersection«/mi»«/mtd»«mtd/»«mtd»«mi mathcolor=¨#0080FF¨»ratio«/mi»«/mtd»«mtd/»«mtd»«mi mathcolor=¨#00B200¨»angle«/mi»«/mtd»«mtd/»«/mtr»«/mtable»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnalign=¨center center center right center left center right center left¨»«mtr»«mtd/»«mtd/»«mtd/»«mtd»«mi»P«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced mathcolor=¨#FF0000¨»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»y«/mi»«/mrow»«/mfenced»«/mtd»«mtd/»«mtd»«mi mathcolor=¨#0080FF¨»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mi mathcolor=¨#00B200¨»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd»«mi mathcolor=¨#0080FF¨»f«/mi»«mfenced mathcolor=¨#0080FF¨»«mo»§#952;«/mo»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo mathcolor=¨#00B200¨»§#952;«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Generally, when a discussion includes the unit circle, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» are used to represent coordinates and the sine function is written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«mo»=«/mo»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», not «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable»«mtr»«mtd»«mtable columnalign=¨right center right center center center¨»«mtr»«mtd»«mi mathcolor=¨#FF0000¨»point«/mi»«mo mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathcolor=¨#FF0000¨»of«/mi»«mo mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathcolor=¨#FF0000¨»intersection«/mi»«/mtd»«mtd/»«mtd»«mi mathcolor=¨#0080FF¨»ratio«/mi»«/mtd»«mtd/»«mtd»«mi mathcolor=¨#00B200¨»angle«/mi»«/mtd»«mtd/»«/mtr»«/mtable»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnalign=¨center center center right center left center right center left¨»«mtr»«mtd/»«mtd/»«mtd/»«mtd»«mi»P«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced mathcolor=¨#FF0000¨»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»y«/mi»«/mrow»«/mfenced»«/mtd»«mtd/»«mtd»«mi mathcolor=¨#0080FF¨»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mi mathcolor=¨#00B200¨»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd»«mi mathcolor=¨#0080FF¨»f«/mi»«mfenced mathcolor=¨#0080FF¨»«mo»§#952;«/mo»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo mathcolor=¨#00B200¨»§#952;«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Generally, when a discussion includes the unit circle, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» are used to represent coordinates and the sine function is written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«mo»=«/mo»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», not «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math».