,---------------------------------------- -- -------------- -- - ---------------------------

Unit 2 - The Trigonometric Functions - Classwork

opposite

..•.. ~-

Given a right triangle with one of the angles named 8, and the sides-of the triangle relative to 8 namedopposite, adjacent, and hypotenuse (picture on the left), we define the 6 trig functions to be:

II R IIThe Basic Trig Definitions Ifl \c..@ tJ () Me ift ~G AlIo ~

. 8 opposite 8 hypotenusesm = the cosecant function: csc = ~"----hypotenuse opposite

8adjacent 8 hypotenuse -

the cosine function: cos = the secant function: see = ~--'-"----Ihypotenuse adjacent

.~ . opposite _. adjacentthe tangent function : tan 8 = . the cotangent function : cot8 = .

. adjacent opposite

the sine function:

Given a right triangle with one of the angles named 8 with 8 in standard position, and the sides of the trianglerelative to 8 named x, y, and r. (picture on the right), we define the -6 trig functions to be:

the secant function:

the sine function: .8 ysm =-r

rthe cosecant function: csc8 = -

y-x

the cosine function: cos 8 = -r

-- rsec8 =-

x

the cotangent function: cot8 = ~y

the tangent function: tan 8 = 2:.x

The Pythagorean theorem ties these variable together: X2 + l = r2

You MUST, MUST, MUST know the above thoroughly, inside and out, backwards and forward, and can never-forget It. It must be part of you. Expect quizzes every day for the immediate future to test whether you knowthese definitions. You will find that if you learn them now, this section will be incredibly easy. If you learnthem and immediately forget them, you will struggle throughout this course.

- .A good way to remember the basic definitions is to remember the terms SOH-CAH-TOA. Sine = Opposite,Hypotenuse, ... Cosine = Adjacent, Hypotenuse .... Tangent = Opposite, Adjacent. For the other trig functions(called the co-functions), Sine goes with Cosecant (S goes with C), Cosine goes with Secant (C goes with S),and the other functions both use the words tangent.

r>. Finally, remember that there is no such thing as sine. Sine doesn't exist by itself. It is sin8 or sina or sinx.Every trig function is a function of an angle. The angle must be present.

2. Basic Trigonometric Functions - 1 - www.mastermathmentoLcol11 - Stu Schwartz

rr=------------------------------------------------------

Example 1) Let P be a point on the terminal side of 8. Draw a picture and find the 6 trig functions of 8.

c) p( 5,2) Jl. ~ ~ 15 -L-I- '2."7.. .: ~ '"Y-=~ 1- ~ '2. d) P(1,7)r;;-;; - _ ~ _ .:L _2m I J't.- \fVf ~

,~1.. Slfv1,~- I\. - vr'l -1:'1, ~e ::~:: ~ \j~~ '1- 5 ,~

()r;Je- 1:..-_2--=-,vl.'1f n: ~ I- J1- -vz.;; ?]f, ~G '" AT= 7

~M."2. 'Olt -x a: ~- §__~ l '~(j=7C._~ _

/\- :JL - - , "'-.-...;..-- -.;

----~-~--T__~-- S I ---~~,~----~=-_+--_r~_r~--~~===='==~--~I-----_:~~----1~e)P(l,l) fL~~I~Jr.'=-f[ '1-~\ ~::\ e)p({i,-fi)JL~~l\fZ)'\lG)<.::3 rx.-=\["i. ~-:::'V7

{ All e- 'i- \ ,fL I .n: ~ , I .r::::;-~V'. ~ _;;-:: ,r.; -= -s: I" M /f:j. -= IV, = \q t- ' ~ VI JL 3 ,3 \I -,• .., "" <.- V"" V V' 0 VL - e V7 ,,).VY\.~-:: jC'="3' I ~Q~ .if ""-0 ..••. -,

~~-=~-~_~I ~ ,~~-n-r AJl..c..,Q-=o-:;;-=V2 vr 'l- v'Z. I f'- '3 3ft'_ ~ L I'" c.ro ~ -:. -;:c ~ '3 ,S us ~ ')(..~ v:i. -.:. T

~ e""- y::: \ -e, \ I ec-t~":: 1:...:; IQuadrant Angles: I 1 ~ ~-= 1- ~'£1 I cot$-"&. 1S-... IT r>;

Jt. n r ~ '.f1Let's examine the trig functions ifpointP is not in the first quadrant. Let's make a chart of the signs ofx,y, andr in all of the quadrants and thus, the signs of the trig functions in those quadrants. (r is always positive)

l..tJelL~ S A-11.. e f f) ~ IIll! GI &L.L

S uYI.- ~ :: 'a:- -=- ...\- - +co.c... e- .Jv - ~-

a) P(3,4)

~~~~eC»0 ~ -= ~ := -+ ~(:-)Mc.f)-

~ f} -= 1- :::--±- = (:-)uto- 'y -

~e- -=')(. IS

10zG0f\. 1'"7

J\.. ~ I-f ::=- -= -I

1- ,$

~~= ~ 8 I ')( !.L-- If' I Ce?t~ ~- ::- e--- 'X to

e.to ~ =- ~.A.1U..e

~G ~;. z,

c.v-te

~e='~'d

~~ ::::1!A-A>

~ - ~ - (_'\»c > +- - J

': ~~'-+III -iA K r;

~ Nt16NT

2. Basic Trigonometric Functions

J.~(} ~ ::: S-l ~ C:-)J'L..~&

CP='8- 'y: -\-- -\--~g«..~ 0./1....- - --r:-

~~ -=- '1:.--=-- H :::-c-)uj'rs '-L. +-

IV C A L t, U L.-\}' ~C)s",JE

- 2 - www.mastermathmentor.com - Sill Schwartz

A good way to remember this is the term: A-S- T-c. It says the quadrants in which the 3 basic trig functions arer=: positive: (All- Sine - Tangent - Cosine) I LIkt; If rH. L. J-rtJ 0 GilliS. /4/{ r: CfJ /...e u ,-U;.S II

When we draw pictures of trig functions in quadrants other than quadrant I, the triangle is always drawn to thex-axis. The angle inside the triangle will be called the reference angle. It is defined at the acute angle formed bythe terminal side of e and the horizontal axis.

;, Example 2) Let P be a point on the terminal side of e. Draw a picture showing the reference angle and find the6 trig functions of e.

'J.- -=- V31.. 2 fi

-::.{3=-T

Example 3) We can be given information about one trig function and ask about the others. Draw a picture.

a) If sine = 12 ,ein quadrant I, find cose and tane. b) If cose = ~,e in quadrant IV, find sine and tan d.13 '2.... 3- = =

1: FL:= \3 1-'\ -=-\'L. A~~3-1?.:'.r ,..--- ( .r=Jd ~='L JL~.3 ~:::t-)~9-Y ~lJo/

~ \ '2.. 13S \.At (1= 13 coc: e -; Ii.

:J~g-:;. 13

I _ II - \-z... c:t;7M\, 0 - ~ CJk (;) =- ...:!-

;J 7 I'L/ LJlLL nen« D/I2.E"c....~1·6I\JS, .

c) If tan e = 3,e in quadrant III, find cose and csci) ..~

r> -~- 3

2. Basic Trigonometric Functions - 3 - www.mastermathmentoLcom - Stu Schwartz

.-----------------------------------------------~---~--~

Example 4) In whatquadrant(s) isAA. ,AI-)

~sec8 < 0 and c<f_~< 0Q:9~m @sTII

11-

a) sin8 ~and c9~~ < 0'!..~\f9 ®~mIT.

5 INc) csc8 < 0 and cos8 > 0:!II. ~@ I ~@

IT

d) all trig functions are negative?

Nr=vSlZ fCt-r u::P~-r\W\::)) .

~ u,J cr, ON:> Ao.e P./...w"Y -S' Po, I rille-

l ~'11 Pc ~'I TII.f6' ,Trig functions of quadrant angles:

The picture below shows quadrant angles: Choose a point for each quadrant angle, determine x, y, and r, anddetermine all six trig functions for those angles: Note that angles can be in degrees or in radians.

" 'It}' 1:-:It r:'7

OO(or0) Point (_I ,Q)

X= \ y=~ r=_I_sinO° = 0 cscO° = L)tJo

cosO° = \ secO° = Itan O"> 0 cotOo= vtJo

900( or ~) Point C~-,~)x=~ y=_I_ r=_'_sin90° = l csc90° = I- -

cos90° = ~ see 90° = UJJ 0

tan 90° = UN 1:1 cot90° = a

270° t9~3;) Point (~,2)x=~ y=l-Il r=_'_sin270° = l-\) csc270° = C:::-l)cos270° = 0 sec270° = UN D- -

tan270° = UN!) cot270° = 0

b) 6tan180° + 3csc270°-2secOo'

a) Ssin90° -12cos180° -::-z, 5 ll') - \2. (- I ) ~ \ 1

-~~J(I) - ,. (0)3 (j) '4 _ l3ll)'1.

t:lo) ". _ ~3 - q - -lo -

c) (4 sin 90° -2cos270° - st -=.~l4 (I') - 1-(0) - t;) 1.-: (1.\ _ s-) 1..=- ,

-6sin~ - 5cot90°d) 2

3cos20 _ (3COSO)2

2. Basic Trigonometric Functions - 4 -

190"-----f----'1r

3,liof

180° [or n] Point (=i,~)X= ~t)y=~ r=_'sin180° = 0 csc180° = U/J~cos1800= l:= \) sec1800= t:\)tan180° = C cot180° = UIJO

www.mastermathmentoLcom - StuSchwartz

Example 5) Calculate the following without looking at the chart above:

Domain and Range of trig functions:

~ Domain: We can take the sine and cosine of any angle. But since tan e = Y and cote = ~, we have to worryx y

about angles where y = 0 or x = O. x = 0 along the y-axis so we cannot take the tangent of 90° or 270°. y = 0along the x-axis so we cannot take the cotangent of 0° or 180°. For the csc function we have to be concemedabout angles where y = 0 (0° or 180°.) and for the see function, we have to be concerned about angles wherex = 0 (90° or 270°).

Range: Since we know that trig functions are based on the picture below and that in any right triangle,

r > x and r > y, r must always be the larger side. So since sine = y = smaller, we find that the range of ther larger .

sine (and cosine) functions must be less than (or equal to) 1. And since csc e = y = larger , the range of ~her smaller

cosecant (and secant) functions must be greater than (or equal to) 1. Since tane = y, we find that there is nox

restriction on the values of the tangent function and cotangent functions. This can .be summarized by the tableon the right:

Special Triangles:

Domain:

sin e :all real numbers

cos e :all real numbers

tan e :e ~ 90°,: e ~ 270°

cscii :e ~ 0°,: e ~ 180°

secn :e ~ 90°,: e ~ 270°

cote: e ~ 0°,: e ~ 180°

Range:

-1s;ys;lor[-I,I]

- 1s; y s; 1 or [-1,1]

all real numbers or (-00,00)y s; -lor y ~ 1 or (-00, -1] U [1,(0)y s; -lor y ~ 1 or (-00, -1] U [1,(0)all real numbers or (-00,-1] U [1,(0)

You must know the relationship of sides in both 30° - 60P- 90° and 45° - 45° - 90° triangles.

GJ·.....'1.... -=:-'2. AI ~{i~ 45'"60 ..Ji

;---.J-3..;...L""""L-....I..~f{3 +tdl J I

. In a 30°'_ 60° - 90°, the ratio of sides is 1- -fi - 2. .

2. Basic Trigonometric Functions - 5 - www.mastermathmentor.com - Stu Schwartz

tJ3' I {[ 7 ..l-_J3_ 5' .rla.<;.-r-I1~"'D~'2E- -'2":.D1B(P' V2 .••.~=t),70 G-T-e.77

So, complete the chart: lJAllr-riIIlG IS I""'T~ t;r:!( L.-W)cJ,? 1/:5';' V# -;'lJ~Ido=/'/5; f3 -= /,7.1r>.

I

cpL.

8 sin8 cos8 tan 8 csc8 sec8 cot8

I 300 (or ~) I V,~,~bh-k-~ D,;17 2- 'Z

2 t.J3 V3 '1'73- 0.£ - ~"'3 1,1~5''2- 2- 3- 3O. sz.

..:45° (or ~) I Vz ~~ e .1D7 {2. 1.'11'iVi--Z ~no7fi~-:Z: I , 'f[ llt111.{ 1 I

t),7JJS

60° (or ~) {5 f &~G !.:'2.V3 I I~ l 1/3- e.t"fo - \f5 //75 1/3 :1 • 2 'l fl ~3" (j ':;77I

/. DG"1- '2..

Determine the len ths of the sides (the signs are waiting for you . E) Find the trig functions of these8 Radians Drawing Reference sin8 cos8 tan 8 csc8 sec8 cot8 I

angle/ \\'.0

~

61~ 'f jfYI~'" $TGl?P 3- - \

120°l.Tr o.'1l 'tL +- c-) -(3 IT"i -'2.. V!

, -a~c\ 3 ,'2,oq,. ~ o.ft..~-~ -().~ -1.13 \.15~ -0.911!31~ V "30'0) 2-

3'1r IIS7' t.u·o '-l S'" ~rt. ..,r Git '-t s- f?{Z. 'J"'2.. -V2.

~ r~

..r L-) -I -\~ 135° _ 079 i -; -O.~(JJ7'1.r) L( 2.3 (., ~ D,~61 1.1.j II.( - /.1-{ lJi ~

Lf e: It. Vi.

314 SlY) 'Ilt- '1 (31C, '10.. .$:"\-\ ~L.~\': _2. -{.,}150° 5" IT -on. ' e

~.+-

-~ -..!.. '="O.S117 '2.. v~ I

5'/36) c 2.~ z "i. ss 'Zoo rs -s.srt -\,13l314 I< 30(.0 -o.fb(..

11rtJ.ILf e JNlIltJ... '1 514 )( <;hA-l.l.o..., _'1.. Vi',P- V 3uG (-) -(,-\ + -2 ~2,~0 , -!C>~ 14"3 -o.a~t730 '" j 3.~(., (31~>( 3Obo -l-O,~

"'2.i§ P. (17 --\'l5"~ \,'1:)

'31lf e I)P yrOIf S- c.'-- Lt ceo: y se« -IT -'1]r( J/t D7q

~

(::-} (-'\ +12~~_) - ~ -0.78 -I -t-Is vr - 9~ '-15' ell. ?i-tl101 -/.414 - I • 'i It.{i..f ,3.,

tf -0- '-f,'71 P Bl~ '/ SntlUL)t. S\e~f -"2. J3r~~if~o I o sz l-'l lr'\ {3+ fs' -1-.•..q ~(}) 311./1 -113 -o.~'~~\ -DS 1.'1381(,,- '/ 2fJ/4C -z.. 1.73 -l.lr.rf ~.

5"2'~ eC\- alli;"f Srnf)./..L. ,. {/1(;Gf -"l.. I

3000 ~o -.,f3~) L+- ) c-J n 7.- '6sl~aJ .J 15:2.3 '"'--i' ~ e.s -V3 -1.llf1314 Y JO(Oo -e.,~'b -1.73 - 0,5"11- if, '71

e~ If selL '-I r' lill '-t see-<,,,

07'1 V4!;C - ().'16'1 ,e\-) -IT \Pi: -I~(~5°

--:- -\7 I{s-) 4 ~.~o Lf5 i:(Z ~~ If-z. 6-,101 - ( I,t.{ ~'-{ I .Lj \41(,,''2~

s~ 3~.$1'I1J\U- SI4 )(. ~hALl.()vJ 1. -{3\~..!n Y (~, - o. 5'"11 '-'2.. J3 I,IS)330° -~ -6:1 ~e,(,.G -r,13/;~()) 19 1;-.1G. \

814,,( '00,",(:) ~-z. -\5¥

The Special (Friendly) Angles' j</J(ft..J vioul{, &UAOfLA-l.1-rnL /)A/C,LG.f ])£CII11I4LL y~. • 1/ ~ z:a::

'Z- -= I.S7 11":-3./'1 '2::: '/; 7/ 'L'II-=- ~,2 8Any multiple of 30°,45° or 60° is considered a special angle (or a quadrant angle) and we can compute trig

. functions of these angles. A) Draw it. B) Establish the quadrant and fill in the signs of the sides rememberingASTC. C) Find the reference angle (which will be 30°,45° or 60°) D) It is one of the special angles above.

angles.\ H13tHLUIiH. 6l0CI<.Sor If -;r -;r(; 3" Y 230c tDo°I.{S'f>'jf)D

r=>;

Jo ,.~ 3JO

I-S'-l-\\(1£S) q ~Lvr<.f,/"

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\-"3- 5"-7'-15" 1"35 2'2!i'" .3((

5,Gr;-f U3 \.13)

51, ~Ll. O'/AJ IT) 0-577514 YJ-,O.f'~~

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Example 6) Calculate each of the following expressions. Do not look 'at the chart on the previous page as your=>; will not have it in an exam. As you did, draw a picture which will help you to calculate the values of trig

expressions. Label the picture in case you have to use it again.

~------------------------~--------+--------------------------------L

a) 8sin300-6cos60° =::: g L~) - t: ( ~) -:: ~ - 3

r1el?l em ~e.e_ '2.- (... I

,t(/)L ~ + C»;; 'I -:.I 1.

"7 -+ Z ::: r

e) -5 sin 90° - 2cos120° =-5 sin 90° + 2cos120°

Co-terminal Angles: t

\1 IIC-z +'1 ,,2.)

So far, our angles have all been between 0° and 360°. What about angles outside that range? We will find thatsince 360° represents one full rotation, that when we take a trig function of an angle greater than 360°, thereference angle is the same as the angle created when subtracted 360° from the original angle. So we can makethis claim. We may add or subtract any multiple of 360° (2rr) to any angle and the trig functions of that angleremain the same. Not that we are not saying the angle remains the same; 100° and 460° are clearly differentangles, but sin l Of)? = sin460°

(!o--r(7lZmJIlltJ-L IlJc,I-GS - (!o~K. PC!lLe w 1/")4 /,~SExample 7) For each angle given, find the angle between 0° and 360° which is co-terminal and then find thesigns of the trig functions of that angle. ,.'LL /J _. tJ A / fl d. fi'

7 IJIJM~G!l2j C-/1£C.K (7.,. (....orvrt./'HI-J,a.(..

() l<l Co-terminal sin() cos()~.

csc()tan() sec() cot()~ angleIL.

K. (between0° and 360°)

400° I ¥OO tf 0 b 6/. 839e> t. .3 o~-Lj 11/9/7-;s(oQ :r: f), 0i27 o. 7{Pfo 0 I, §"§57-c:850° f.ro J! l,Je 0 f).7~b() -(j,t'f27 . - 1.5;-r;7 -fJ.e39o-; 'Z-O - /./9/7 /,J o 5't.j7" .3e>1275° ,'2. 7S"

IlL/9SfO -0.9b59 o. l(,79IO.!'O -6, 2 5~f -3,8(;,37 -I,()'352 3,7J201'1:7

-231° 13'0-' 11-/39° 0.717/ - O.~2q3 - a ~& 9'2. I,2§ fa7 -/.5'990 -I./~o3-721 ° 112D- :r: ,0 (J, ()/7'f tJ,997'd' o. 0/7,/ r7. 2. rf' f,OOOI 5'7. 2.f'11171t /11- Q{<' UL !J---n6 - o.~'c-0 (J, S -/,732.0 '2. =t), 5" 773

3 5 :3 30>/ -}.1S"'17

2. Basic Trigonometric Functions

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Unit 2 - The Trigonometric Functions - Homework

1. Let P be a point on the terminal side of e. Draw a picture showing the reference angle and find the 6 trigfunctions of e.

U-c.e ~ c vz:)t-~)

~e-~ I

a) If tanf = ~,e in quadrant I, find cose and sine.3

.t-.':. ~ + - +QJf/'~ ) ~

J/..., -:: 5"

b) If cose = .J3,e in quadran@V,findsineandtan8.2 ~) ~)'"'L~03 /L:::.'2.. 1'J\-:::~L\-3 :::\~ -:..~\)

I-I) L l z: ) -Vis.un ~ = \:::2::. \WYl f}- ~ \. 1[$ Q ~ 3""2. Basic Trigonometric Functions - 8 - www.mastermathmentor.com - Stu Schwartz

.---------.-------------------------~-----~-------------

5 ® 5 . ~ .c) If sinO = -,0 in quadrant II find see 0 and cot O. d) If secO = - -,0 in quadran~ find sin 0 and tan O.8 Ie) E::-) 2 r (-) 8-)

~ '=.;; JL ~ ~ \~\:: \J L'4 - L ~ ::- V34 JL -=-? "1. "=' \.-:- '2.) ~ -= - \J '25"" - q -;::..- VzT'K -:: ~~3q~

Jl. ~ -~~ J -S"~ (\tfYl''\ -= - m -6t- e::. -_~ -=- ~~&.". --:;.- ~r~- ~t}""_'=3aO(l. 3Q 0- tl ~ L- ~"I.... -\]30:' -SG. '{s""

----------------------------~--------.~~- ._---

.f) If cosf = -fi and~ find sin 0 and t~O.1. -.IiI o (L Iil t-:-) <::- )I. ~l'L SSI

. e) Iftan(L.,:- 5,0 in quadran@ find sin 0 and see O.- 8 ~

~e.u ~"'- 0 1C= I JL -::.~ \1.~sr"\"::~

I.

a) sinO> 0 and cosO <:0I 4II rr -zn.

b) cscO > 0 and cotO < 0::t:.-;IL TTJ"N

1I11d) cscO < 0 and cosO < 0

N1nc Tr:rlITc) secO < 0 and tanf < 0

IT!f III IT+JS[

-.lLL4. Find the value of the following (do not look at the chart - make a small picture and calculate the values)

a) 5sin90° -7cos180°.::;

::: ~(\) - '1 e.-I) ~ S+1 = 1'2..b) 4secO° + 7csc270°-;::

::;. ~ C\ I 4- "1(-I) :: L\ -"1 :: C: 3)

W (?1L..l ~ G-T rY) e (l.-l:- I-J T Q ,.-41 sLV-I-~Yt.

-e-~-cl-~)-:-Osc~)Oo~C(S~lr:roo;----···--:--r~((in:~o~-lsec)?~~2~~ecoO)=:::(::1) .. I~ l-2)(())=0

2. Basic Trigonometric Functions - 9 - www.mastermathmentor.col11 - Stu Schwartz

-5. For each statement, determine whether or not it is Possible (P) or Impossible (1).

a) sine = -5 I. Arlc... - \ s, Wl.e ~ Ic) 2cose+5.5=4 r J..r(<:- CAbS:: -I:;'e) cscrz + sinf3 =.5 r 0 + -~) :0:. 0.5

P k-/ft k ft :::1. ,7Cj1:. k/ e. I -+- I :: 2 m f/-x.,

f Jv(L. .s&n9o +~$ ::-Z

b) tane + 1= 3.79d) sinrz + cotf3 = 8f) sin e» cosf3 = 2

6. Find the value of the following (do not look at the chart - make a small picture and calculate the values)

a) 6sin300-4cos150°;:=-

-=-<0 C~) - ~ (-~J-:::.3+-1..V3

b) 8sin600-4sin300°:.

-=- ~ (~) - ~ ~V}) -= l:ff-= r.a V3

------------'----------------t---------- _c) (4 tan 120°)( 8cos225°) =- d) 6 sin 3150 + 8 tan135° ::.

- Y (- \13)l~-~) ~-~+ ~ (- II-= \ b ~ -= (:: 6) - 3 vz:

-e)-8 c-sc-3-00--_--g -rr-'l. ):---_-,--, 10-'r-::;--3--2.,-----+---------·-------\ \. 'I ~ _ f) -2cos225°-4cot315°+3 =:-

cot 330° - - \f3 - :5 \-:. - 2 t~ )- ~(-/)+ 3

g) sin" 225° - cos' 225°-=.

~ rk)~- C~)'l=- 0

~h:Ho'2..'7 <:l

- 30 -2'J 330 ".

( )

'4-------

. 2 3n 2 Tr:J) cOS"4 - csc (5

I'2!iT

th t7 F h fed t . 1 d th . fth t fi t fth tor eac va ue 0 , e ermme e co- ermma ang e an e Signs 0 e ng unc IOns 0 a ang e.e Co-terminal sine cose tane csce seci) cote

angle(between

0° and 3(;iOO)700° (120) 3Lftl 0 - o,y/zo 0,939' - -o.3~J'l -2, ~2J~ t. 0 (, <f I -'L 7'-17¥1525° (It{'-/<o) 85"° 0, 99(, / e. o B7/ II. '-/Jeo /. ~e 3& II. ({737 &.o~ 7'-/-485° -I- 7Z-<> -11,),=;- l'3~o -o.fJ/9/ -0. b735" I. i.I'l 91 -1.~Z.07 - /.7'(3Y- ().760 'Z.2.5n '2.11 0,5"1( =-) 90° / 0 UNO J V!J ~ (5)

-20;( ItO'205"'. ""11.f~ ~d. '"/318 -c>,9009 fJ.1~/~ -'2..30i7 - /./099 2.o7('r---7 It"

-3'"O()Cl o"I - ~I'f, 1.t e~T

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C ~ t, l.. IJ I".A-,\ tll'<. •

- 10- www.mastermathmentor.com - Stu Schwartz