1-1 Introduction to Trigonometry

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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θ 7 !


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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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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

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1-1 Introduction to Trigonometry