L2.2 A2 Trigonometric Ratios - Part 1
Completion requirements
Unit 2
Trigonometry
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Part 2.2A corresponds to section 4.3, starting on page 191 of your Pre-Calculus 12 textbook.
The Primary Trigonometric Ratios
In Math 10C, the sine ratio was defined as “the ratio of the side length opposite an angle to the length of the hypotenuse in a right triangle”. This was often accompanied by a diagram similar to the one shown.
Using the unit circle to define the sine ratio will keep this original
definition intact, but will allow for the sine to be defined for angles
less than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» and greater than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»90«/mn»«mo»§#176;«/mo»«/mstyle»«/math». Take another look at an angle, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math», drawn in
standard position on the unit circle.
So, using this diagram
Because the unit circle always has a radius of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», this expression will always simplify to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math».
The same type of reasoning can be used to show that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». Start with the Math 10C definition of cosine as “the ratio of the side length adjacent to an angle to the length of the hypotenuse in a right triangle”, and note that when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math» is drawn in standard position, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» is the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinate of the point of intersection of the terminal arm and the unit circle.
Similarly, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»tan«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» can be defined using the point of intersection of the terminal arm and the unit circle.
To summarize:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi mathcolor=¨#FF0000¨»y«/mi»«/mrow»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»cos«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi mathcolor=¨#0080FF¨»x«/mi»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»tan«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mi mathcolor=¨#FF0000¨»y«/mi»«mi mathcolor=¨#0080FF¨»x«/mi»«/mfrac»«/mstyle»«/math»
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mi»y«/mi»«mn»1«/mn»«/mfrac»«/mrow»«/mstyle»«/math»
Because the unit circle always has a radius of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», this expression will always simplify to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math».
So, when «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«/mstyle»«/math» is drawn in standard
position, «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi
mathcolor=¨#FFFFFF¨»sin«/mi»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«/mstyle»«/math» is the «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi
mathcolor=¨#FFFFFF¨»y«/mi»«/mstyle»«/math»-coordinate of the
point of intersection of the terminal arm and the unit circle.
The same type of reasoning can be used to show that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». Start with the Math 10C definition of cosine as “the ratio of the side length adjacent to an angle to the length of the hypotenuse in a right triangle”, and note that when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math» is drawn in standard position, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» is the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinate of the point of intersection of the terminal arm and the unit circle.
When «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«/mstyle»«/math» is drawn in standard
position, «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi mathcolor=¨#FFFFFF¨»cos«/mi»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«/mrow»«/mstyle»«/math» is the «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi
mathcolor=¨#FFFFFF¨»x«/mi»«/mstyle»«/math»-coordinate of the point of
intersection of the terminal arm and the unit circle.
Similarly, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»tan«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» can be defined using the point of intersection of the terminal arm and the unit circle.
When «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«/mstyle»«/math» is drawn in standard
position, «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi
mathcolor=¨#FFFFFF¨»tan«/mi»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«/mstyle»«/math» is the «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi
mathcolor=¨#FFFFFF¨»y«/mi»«/mstyle»«/math»-coordinate divided by the
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi
mathcolor=¨#FFFFFF¨»x«/mi»«/mstyle»«/math»-coordinate of the point of
intersection of the terminal arm and the
unit circle, or simply «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi mathcolor=¨#FFFFFF¨»tan«/mi»«mo
mathcolor=¨#FFFFFF¨»§#952;«/mo»«mo mathcolor=¨#FFFFFF¨»=«/mo»«mfrac
mathcolor=¨#FFFFFF¨»«mi»y«/mi»«mi»x«/mi»«/mfrac»«/mrow»«/mstyle»«/math».
To summarize:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi mathcolor=¨#FF0000¨»y«/mi»«/mrow»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»cos«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi mathcolor=¨#0080FF¨»x«/mi»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»tan«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mi mathcolor=¨#FF0000¨»y«/mi»«mi mathcolor=¨#0080FF¨»x«/mi»«/mfrac»«/mstyle»«/math»