Module'4'Lesson'3:''The'Unit'Circle'
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The!Unit!Circle!is!the!most!important!concept!you!will!learn!in!these!Trigonometry!units.!!
If!you!plan!to!take!any!level!of!Calculus!in!high!school!or!college,!then!you!will!definitely!
benefit!from!being!familiar!with!how!the!Unit!Circle!works.!!!!
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Recall!what!you!know!about!the!Unit!Circle!thus!far.!!!
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The!Unit!Circle!is!a!circle!with!its!Center!at!the!Origin!(0,!0)!and!has!a!radius!equal!to!1.!!
The!equation!for!the!Unit!Circle!is!! ! + ! ! = 1.!!Since!the!circle!is!centered!at!the!origin!
and!has!a!radius!of!1,!the!x"and!y!intercepts!are!shown!above.!!The!xGintercepts!are!(G1,!
0)!and!(1,!0);!the!yGintercepts!are!(0,!G1)!and!(0,!1).!
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Radian'Measures'
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In!this!module,!will!just!be!focusing!on!the!most!common!radian!values!associated!with!
the!Unit!Circle.!!We!will!look!at!positive!and!negative!angles!that!are!between!0!and!2π!
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radians.!!The!most!common!angles!are!multiples!of! , , !"# .!
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Multiples'of'90°'or'π/2'radians'
Look!at!our!Unit!Circle!as!multiples!of!90°!angles!(with!initial!side!on!positive!xGaxis).!!!
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Notice!the!angles!in!radian!measure!are!also!stated.!!!Recall:!!180!degrees!=!π!radians.!
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GOAL'"'Try'to'understand'and'memorize'where'the'angles'with'the'given'radian'
measures'are'located'and'their'corresponding'ordered'pairs'(x,'y).!
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Note:!!We!could!also!use!negative!angle!measures.!!Recall!moving!in!the!clockwise!
direction.!
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Multiples'of'45°'or'π/4'radians'
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Since!we!just!learned!the!ordered!pairs!for!90°,!180°,!and!270°,!we!will!only!focus!on!
angles!45°,!135°,!215°,!and!315°.!!We!will!look!at!the!radian!measures!of!these!angles!
and!their!corresponding!ordered!pairs.!
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Similarly,!we!can!also!consider!negative!angles!that!are!multiples!of!45°! !"! ! 4 !as!we!
did!with!multiples!of!90°! !"! ! 2 !!as!we!did!on!the!previous!page.!
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These!values!come!from!the!relationship!of!sides!in!a!45°G45°G90°!right!triangle.!!See!the!
figure!below.!
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Multiples'of'60°'or'π/3'radians'
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Notice!the!multiples!of!60°!below!and!their!corresponding!radian!measures.!
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We!have!already!learned!the!ordered!pairs!for!180°!and!360°.!!The!ordered!pairs!for!the!
remaining!angles!are!as!follows:!
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Similarly,!we!can!also!consider!negative!angles!that!are!multiples!of!60°! !"! ! 3 !as!we!
did!with!multiples!of!90°! !"! ! 2 !!as!we!did!on!page!2.!
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These!values!come!from!the!relationship!of!sides!in!a!30°G60°G90°!right!triangle.!!See!the!
figure!below.!
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Multiples'of'30°'or'π/6'radians'
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Notice!the!multiples!of!30°!below!and!their!corresponding!radian!measures.!
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We!have!already!learned!the!ordered!pairs!for!60°,!90°,!120°,!180°,!240°,!270°,!300°!
and!360°.!!The!ordered!pairs!for!the!remaining!angles!are!as!follows:!
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Similarly,!we!can!also!consider!negative!angles!that!are!multiples!of!30°! !"! ! 6 !as!we!
did!with!multiples!of!90°! !"! ! 2 !!as!we!did!on!page!2.!
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These!values!come!from!the!relationship!of!sides!in!a!30°G60°G90°!right!triangle.!!See!the!
figure!below.!
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The'Unit'Circle'
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Combining!all!of!these!radian!values!with!their!ordered!pairs!gives!us!the!following!
figure.!!Note!that!radian!measures!are!given.!!Refer!back!to!previous!figures!for!degree!
measures.!!The!colors!for!each!angles!are!included!so!that!you!can!see!the!reflective!
properties!of!the!ordered!pairs.!!
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You!can!find!the!Unit!Circles!on!the!web.!!Just!search!for!“Unit!Circle”!and!find!a!nice!
printable!version.!
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Don’t!be!overwhelmed!just!yet.!!If!you!break!it!down!and!study!the!“multiples!of!π/2,!
π/4,!π/3,!and!π/6”!separately,!you!will!eventually!get!the!hang!of!it.!!The!Unit!Circle!
takes!practice!memorizing.!!You!can!find!numerous!videos!and!tutorials!online!to!help!
you!memorize!the!ordered!pairs.!
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Helpful'Memorization'Tool'
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Here!is!one!way!to!help!you!memorize!the!ordered!pairs.!!!
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We!will!only!look!at!Quadrant!I.!!The!ordered!pairs!for!0!and!π/2!radians!should!be!quite!
simple.!!!
Next,!let!each!x!and!y!coordinate!have!a!denominator!of!2.!
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Starting!at!the!top!and!working!your!way!down!to!the!xGaxis,!label!the!x!coordinates!1,!2,!
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Starting!at!the!top!and!working!your!way!down!to!the!xGaxis,!label!the!y!coordinates!3,!
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Next,!take!the!square!root!of!all!of!your!numerators.!
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We!can!simplify! 1 = 1,!and!we’re!done!!
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Looking!back!at!the!complete!Unit!Circle,!you’ll!notice!that!each!of!the!following!
Quadrants!are!simply!“mirror!images”!or!“reflections”!of!Quadrant!I.!!!
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More!specifically,!all!angles!that!are!multiples!of!π/4!have!the!same!ordered!pair!values,!
except!some!values!may!be!positive!or!negative!depending!on!what!Quadrant!the!
terminal!side!of!the!angle!lies!in.!!The!same!follows!for!π/3!and!π/6.!
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Take!a!look!at!which!angles!are!positive!and!negative!in!each!Quadrant.!
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Try'your'hardest'to'start'memorizing'the'Unit'Circle'now!''It'will'help'you'in'the'
future!'
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