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8/9/2019 1-1 Introduction to Trigonometry
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Trigonometry
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Vertex – the endpoint of the ray.
Vocabulary:
Angle – created by rotating a ray about its endpoint.
Initial Side – the starting position of the ray.
Terminal Side – the position of the ray after rotation.
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Initial side
I n i t i
a l s i d e
Vertex
Vertex
T e r m i n
a l s i d
e
Terminal side
This arrow meansthat the rotation
was in acounterclockwisedirection.
This arrowmeansthat therotationwas in aclockwisedirection.
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Positive Angles – angles generated by acounterclockwise rotation. Negative Angles – angles generated by a clockwiserotation. We label angles in trigonometry by using the reek
alphabet.α ! reek letter alphaβ ! reek letter betaφ ! reek letter phiθ ! reek letter theta
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Standard Position – an angle is in standardposition when its initial side rests on the positi"ehalf of the x!axis.
#ositi"e angle in standard position
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There are two ways to measure angles$
%egrees
&adians
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%egrees :•
There are '() ° in a complete circle.• * ° is *+'() th of a rotation.
&adians:• There are , π radians in a complete circle.• * radian is the si-e of the central angle when the
radius of the circle is the same si-e as the arc of
the central angle.
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Coterminal angles – two angles that share acommon "ertex a common initial side and acommon terminal side.
/xamples of 0oterminal 1ngles
α
β
α and β are coterminal
angles because they sharethe same initial side andsame terminal side.
0oterminal angles couldgo in opposite directions.
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/xamples of 0oterminal 1ngles
α and β are coterminal
angles because they sharethe same initial side andsame terminal side.
0oterminal angles couldgo in the same directionwith multiple rotations.
α
β
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/xample :4ind two coterminal angles 5one positi"e and onenegati"e6 for the following angles.
θ 7 ,8 °
positive coterminalangle : ,8 9 '() 7 ' 8 ° negative coterminalangle :
,8 – '() 7 ! ''8 °
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/xample :4ind two coterminal angles 5one positi"e and onenegati"e6 for the following angles.
θ 7 !
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/xample :4ind two coterminal angles 5one positi"e and onenegati"e6 for the following angles.
θ 7 π+;
positive coterminal angle : π+; 9 , π 7 π+; 9 *= π+; 7 *8 π+; rad
negative coterminal angle : π+; ! , π 7 π+; ! *= π+; 7 !*' π+; rad
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/xample :4ind two coterminal angles 5one positi"e and onenegati"e6 for the following angles.
θ 7 != π+<
positive coterminal angle :!= π+< 9, π 7 !=π+< 9 * π+< 7 *= π+< rad
negative coterminal angle :!= π+< !, π 7 !=π+< ! * π+< 7 !,, π+< rad
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Complementary angles – two positive angleswhose sum is
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/xample :4ind the complement of the following angles if oneexists. θ 7 ,< °
complement 7
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Supplementary angles – two positive angleswhose sum is * ) ° or two positive angles whose
sum is π.
To 3nd the supplement of a gi"en angle you
subtract the gi"en angle from * ) ° 5if the anglepro"ided is in degrees6 or from π 5if the anglepro"ided is in radians6.
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/xample :4ind the supplement of the following angles if oneexists. θ 7 ,< °
supplement 7 * ) – ,< 7 *8* °
θ 7 *); °supplement 7 * ) – *); 7 ;' °
θ 7 π+8supplement 7 π! π+8 7 8 π+8 ! π+8 7 =π+
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&anually Converting from +egrees to)adians :?ultiply the gi"en degrees by π radians+* ) °
/xample :0on"ert the following degrees to radians
*'8 °
' π radians =
*'8 degrees π radians 7 * * ) degrees
*'8 π radians 7 * )
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&anually Converting from )adians to+egrees :?ultiply the gi"en radians by * ) °+π radians
/xample :0on"ert the following radians to degrees.
!π+' radians
!() °
!π radians * ) degrees 7 ' π radians
!* ) π degrees 7 ' π
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?ultiply the gi"en radians by * ) °+π radians
/xample :0on"ert the following radians to degrees.
,
≈**=.8< °
, radians * ) degrees 7
* π radians
'() degrees 7 , π
5if you don@t see the degree
symbol then the anglemeasure is automaticallybelie"ed to be a radian.6