,9troduc;n & ,review

    ,a ,f/ ,glimpse ( ,calculus
  ,9 hi< s*ool1 ,i didn't /udy calculus1
b ,i rememb] r1d+ ab x 9 ,5cyclopa$ia
,britannica4 ,:ile ,i didn't "u/& a lot
( :at ,i r1d1 ,i rememb] 4c]n+ #3 g5]al
facts3
  =1_4 ,calculus 9volves -put+ ! slope (
curves4
  =2_4 ,calculus 9volves -put+ ! >1s "u
curves4
  =3_4 ,calculus 9volves no;ns f physics
l veloc;y & a3el],n4
  ,at f/ gl.e1 ^! 3cepts seem -pletely
unrelat$4 ,%dn't ,calculus re,y 2 #2 or
#3 sep>ate subjects8 ,! answ] turns \
6be no4 ,calculus is re,y "o subject1 &
"o (! c5tral f1tures ( t subject is !
deep 9t]connect$;s (! #3 id1s ( slope1
>ea1 & dynamical *ange4 ,explor+ ?
3nec;n is "o (! ma9 objectives ( ?
c\rse1 b let me />t 0try+ 6%[ y ": we're
h1d+4
                                      #j
  ,let's />t )! simple/ possi# curve1 a
horizontal /rai<t l9e1 say1 y .k #65,
%[n 9 ,figure #1_4
  ,suppose we want 6-pute ! >ea "u ?
graph4 ,we c't m1n ! total >ea 2t
t .k -,= & t .k ,=, s9ce t wd 2 9f9ite4
,9/1d1 we m/ m1n ! >ea "u ! graph & abv
! ;t-axis 2t1 say1 t .k #0 & t .k x_4 ,=
e *oice ( ;x we get a di6]5t >ea2 ! >ea
2+ ! >ea (a rectangle ) hei<t #65 & wid?
;x_4 ,9 o!r ^ws1 ,a .k #65x_4
  ,n[ ? >ea c 2 reg>d$ z a func;n 9 xs
[n "r1 & plott$ l ,figure #2_4
  ,at ? /age1 we've solv$ ! pro#m ( f9d+
! >ea "u \r v simple curve1 "o (! #3
pro#ms ,i sd calculus 0 ab4 ,n[ "h -es !
nice 4cov]y4 ,suppose we 7 6look at !
second func;n1 ! "o plott$ 9 ,figure #2,
& 6ask = xs slope4 ,:at wd we get8 ,!
slope is #65, &! slope doesn't *ange4
,x's alw ! same at e po9t4 ,s if we 7
6plot ! slope1 we'd get a 3/ant func;n
y .k #65 %[n 9 ,figure #3_4

                                      #a
  ,h y "e se5 a plot l ? 2f8 ,( c\rse3
x's ! orig9al func;n 9 ,figure #1_6 ,s
:at we seem 6see 9 ? example is t !
pro#m ( -put+ ! slope (a func;n &! pro#m
( -put+ ! >ea "u a graph >5't #2 di6]5t
pro#ms - !y're ! same pro#m d"o 9 di6]5t
direc;ns4 ,! >ea pro#m l1ds u f ,figure
#1 6,figure #2, &! slope pro#m l1ds u f
,figure #2 back 6,figure #1_4
  ,? isn't j a co9cid;e ( pick+ a v
"picul> func;n 62g9 )4 ,x "ws 9 g5]al1 &
x is "o way ( /at+ a v important result
call$ ! ,funda;tal ,!orem ( ,calculus4
  ,s n[ we h a 3nec;n 2t ! pro#m (
slopes &! pro#m ( >1s4 ,:at's ! 3nec;n
6physics8 ,well1 suppose ! orig9al
func;n y .k #65 repres5t$ a veloc;y---
"s"o driv+ at a 3/ant #65 mi_/;h4 ,:at
does ! >ea func;n y .k #65t repres5t8
,x's ! total 4t.e travell$1 />t+ at "t
t .k #0_4 ,put ano!r way1 x's ! posi;n
(! c> if xs />t+ posi;n is #0_4 ,s :5 we
take ! slope (! posi;n func;n1 we get !
veloc;y func;n1 & :5 we take ! >ea "u !
                                      #b
veloc;y func;n1 we get ! posi;n func;n4
,s9ce 9 physics1 "o is (t5 mov+ back &
=? 2t posi;n & veloc;y1 calculus 2g9s
6feel l a use;l tool = do+ physics1 too4

,a second example
  ,:at if we 7 6/>t )a m -plicat$ func;n
?an a horizontal /rai<t l9e8 ,! next
simple/ func;n wd 2 a slant$ l9e "? !
orig91 l y .k #2t, : is %[n 9 ,figure
#4_4
  ,! >ea "u ? curve 2t t .k #0 & t .k x
is ! >ea (a triangle ) base ;x & hei<t
#2x_4 ,? >ea is ,a .k x^2, : is plott$ z
,figure #5_4
  ,n[ ?+s get tricky4 ,we ?9k t if we
/>t ) y .k #2t, take ! >ea 6get
  ,a .k x^2, & !n take ! slope ( ? new
curve1 we w get back 6": we />t$4 ,t is1
! slope (! p>abola ,a .k x^2 cle>ly
di6]s f po9t 6po9t1 b we ?9k t at any
"picul> po9t x .k t, ! slope (! curve
,a .k x^2 w turn \ 6be #2t_4 ,is ?
r1sona#8
                                      #c
  ,qualitatively1 x is c]ta9ly plausi#4
,! slope (! p>abola at x .k t t is
positive = positive t, negative =
negative t, & #0 = t .k #0_4 ,! value (
y .k #2t (xs hei<t abv ! ;x-axis) is al
positive1 negative1 or #0 ac z ;t is
positive1 negative1 or #0_4 ,fur!r1 x is
obvi\s f symmetry t ! slope (! p>abola
at x .k -t is ! negative (! slope at
x .k t_2 ! hei<t ( y .k #2t has ? same
symmetry4 ,s ! slope ( ,a .k x^2 at
x .k t is giv5 ei by y .k #2t or 0"s v
closely relat$ func;n4
  ,6get a quantitative e/imate ( ^!
slopes1 we %d probably pick a "picul>
*oice ( t, say1 t .k #1_4 ,figure #6 %[s
a picture (! p>abola ne> t .k #1_4
  ,) ? p>abola1 ,i've drawn #3 l9es4 ,!
solid curve is ! tang5t l9e1 ^: slope we
don't "k h[ 6-pute4 ,! #2 da%$ l9es >e
secant l9es1 l9es t 9t]sect ! curve at
#2 po9ts4 ,we c easily -pute _! slopes4
  ,! /eep] da%$ l9e 9t]sects ! p>abola
at ! po9ts (1, 1) & (2, 4)_4 ,xs slope
                                      #d
is "!=e ?4-1_/2-1# .k #3_4 ,x is /eep]
?an ! tang5t l9e4
  ,! %all[] da%$ l9e 9t]sects ! p>abola
at ! po9ts (0, 0) & (1, 1)_4 ,xs slope
is "!=e ?1-0_/1-0# .k #1_4 ,x is n z
/eep z ! tang5t l9e4
  ,! slope (! tang5t l9e m/ lie "s": 2t
! slopes (! #2 da%$ secant l9es4 ,"s":
2t #1 & #3 mi<t 21 maybe1 #2_8 ,& 9
fact1 ? is ! slope we 7 expect+1 s9ce \r
3jecture has be5 t ! slope (! tang5t l9e
to ,a .k x^2 at ! po9t x .k t t is #2t_4
  ,x is wor? reflect+ on h[ we cd get a
bett] approxim,n 6! slope (! tang5t l9e4
,x is al wor? try+ ! same calcul,n at
po9ts o!r ?an x .k #1_4

    ,basic ,algebra ,review
  ,? is a qk rem9d] ( "s (! basics (
algebra y mi<t f9d yrf ne$+ 6"k4 ,i make
no claim ( -preh5sive;s4
  ,! ,py?agor1n ,!orem implies t ! 4t.e
2t ! po9ts (x0, y;o") & (x, y) is
  >(x-x0)^2"+(y-y0)^2"]_4
                                      #e
  ,! slope (a /rai<t l9e is .,dy_/.,dx,
": .,dy is ! di6];e 9 ! ;y coord9ates (
#2 po9ts on ! l9e1 & .,dx is ! di6];e 9
_! ;x coord9ates4
  ,! /rai<t l9e ) slope ;m & ;y-9t]cept
;b is y .k mx+b_4
  ,! l9e "? ! po9t (x0, y0) & hav+ slope
;m is
  ?y-y0_/x-x0# .k m,
  i4e41 y-y0 .k m(x-x0)_4
  ,! l9e "? ! two po9ts (x0, y0) &
(x1, y1) has slope
  m .k ?.,dy_/.,dx# .k ?y1-y0_/x1-x0#_4
  ,xs equ,n is "!=e
  y-y0 .k (?y1-y0_/x1-x0#)(x-x0)_4
  ,example3 ,! l9e "? ! po9ts (1, 3) &
(5, 14) has equ,n
  y-3 .k ?14-3_/5-1#(x-1)_4
  ,! circle ( radius ;r c5t]$ at
(x0, y0) 3si/s ( all po9ts (x, y) ^:
4t.e f (x0, y0) is ;r_4 ,? m1ns ! circle
has equ,n
  >(x-x0)^2"+(y-y0)^2"] .k r,

                                      #f
  or1 m simply1
  (x-x0)^2"+(y-y0)^2 .k r^2_4
  ,a circle ( radius ;r has >ea .pr^2 &
circumf];e #2.pr_4
  ,! roots (! equ,n ax^2"+bx+c .k #0 >e
  x .k ?-b+->b^2"-4ac]_/2a#_4
  ,e "o ( ^! facts is wor? memoriz+4

    ,trigonometry ,review
  ,! trig func;ns >e func;ns t take
angles z >gu;ts1 & return numb]s z
values4
  ,9 calculus1 z "?\t ma!matics1 angles
>e alm alw describ$ 9 radians4 ,! radian
m1sure ( an angle is ! l5g? (! >c on !
unit circle 3ta9$ )9 ! angle1 z %[n 9
,figure #7_4
  ,! circumf];e (! unit circle is #2.p,
s a #360 angle m1sures #2.p radians4
,o!r radian m1sures p (t5 rememb] >e
  #180 .k .p rad
  #90 .k ?.p_/2# rad
  #45 .k ?.p_/4# rad
  #30 .k ?.p_/6# rad
                                      #g
  #60 .k ?.p_/3# rad
  #120 .k ?2.p_/3# rad
  ,63v]t f radians 6degrees1 y "!=e
multiply by #180_/.p_4 ,63v]t f degrees
6radians1 multiply by .p_/180_4 ,af all1
! units c.el nicely 9 a calcul,n l
  #37 .k #37@*?.p rad_/180#
    .k #0.645 77 rad4
  ,! #2 basic trig func;ns1 cos9e & s9e1
>e def9$ z ! ;x & ;y coord9ates1
respectively1 ( po9ts on ! unit circle4
,t is1 ! po9t (x, y) %[n 9 ,figure #8 on
! unit circle at an angle .? f ! ;x axis
has coord9ates x .k cos .?, y .k sin .?
_4 ,x cd h be5 writt5 z
(x, y) .k (cos .?, sin .?)_4
  ,6plot ! trig func;ns z func;ns ( .?,
"o wants ! horizontal axis 6be .?, & "o
wants ! v]tical axis 6%[ ! value (! trig
func;n at angle .?_4 ,if y walk >.d !
unit circle />t+ on ! ;x axis at
.? .k #0, !n y f9d t
  ,:5 .? .k #0, ! ;x coord9ate (! po9t
on ! unit circle is #1_2 s cos #0 .k #1
                                      #h
_4
  ,:5 .? .k ?.p_/2# .k #90, ! ;x
coord9ate (! po9t on ! unit circle is
#0_2 s cos ?.p_/2# .k #0_4
  ,:5 .? .k .p .k #180, ! ;x coord9ate
(! po9t on ! unit circle is -#1_2 s
cos .p .k -#1_4
  ,:5 .? .k ?3.p_/2# .k #270, ! ;x
coord9ate (! po9t on ! unit circle is
#0_2 s cos ?3.p_/2# .k #0_4
  ,:5 .? .k #2.p .k #360, ! ;x coord9ate
(! po9t on ! unit circle is #1_2 s
cos #2.p .k #1_4 ,at ? po9t1 we >e back
6\r />t+ po9t on ! circle2 = l>g] values
( .?, ! values (! cos9e func;n w j
rep1t4
  ,at ? /age1 we "k t ! po9ts (0, 1),
(?.p_/2#, 0), (.p, -1), (?3.p_/2#, 0),
(2.p, 1) >e on ! graph (! cos9e func;n4
,! full graph looks "s?+ l ,figure #9_4
  ,do+ ! same ?+ ) sin .? produces !
graph 9 ,figure #10_4


                                      #i
,two ,rem>ks on ,not,n
  ,/&>d ma? not,n is m ambigu\s &
3text-s5sitive ?an "s p r1lize4 ,s9e &
cos9e >e func;ns1 s 9 ! same way t we
write f(x), we re,y "\ 6write sin (.?) &
cos (.?)_4 ,t way1 x wd look l we 7
apply+ a func;n1 n multiply+ quantities4
,maple requires u 6use ? not,n1 b \tside
,maple1 x is univ]sal practice 6l1ve \ !
p>5!ses4 ,x is al misl1d+4
  ,! not,n sin^2 .? is ev5 m 3fus+4
,does x m1n sin (sin (.?)) or :at8 ,no1
9 fact1 x m1ns (sin (.?))^2_4 ,t is1 x
m1ns 6/>t )! angle .?, take ! s9e 6get a
numb]1 & !n squ>e ? numb]4 ,ag1 ,maple
m&ates ! not,n (sin (.?))^2 r ?an !
%orth& sin^2 .?_4 ,i'll use ! univ]sal
%orth&1 b y %d 2 sure y "u/& :at x m1ns4

,basic ,trig ,id5tities
  ,! unit circle 3si/s ( all po9ts t >e
a 4t.e #1 f ! orig91 i4e41 ( all po9ts
(x, y) satisfy+ >x^2"+y^2"] .k #1_4
,squ>+ bo? sides ( ? equ,n gives a
                                     #aj
simpl] equ,n =! circle3 x^2"+y^2 .k #1_4
,s9ce ! po9t on ! unit circle at angle
.? has x .k cos .? & y .k sin .?, ?
equ,n immly implies ! mo/ well-"kn (!
trig id5tities1 ! ,py?agor1n id5t;y
  sin^2 .?+cos^2 .? .k #1_4
  ,two m id5tities >e obvi\s ei f look+
at ! graphs 9 ,figures #9 & #10 or f
?9k+ ab ! unit circle 2l 9 ,figure #11_4
  ,! po9t ;,p 9 ? figure cd 2 writt5 ei
z (cos (-.?), sin (-.?)) or z
(x, -y) .k (cos .?, -sin .?)_4 ,? gives
u ! id5tities
  cos (-.?) .k cos (.?)
  sin (-.?) .k -sin (.?)_4
  ,^! id5tities (f] an opportun;y
69troduce a pair ( g5]al 3cepts4 ,a
func;n ;f = : f(-x) .k f(x) = all ;x is
call$ an .ev5 func;n4 ,a func;n ;g = :
g(-x) .k -g(x) = all ;x is call$ an .odd
func;n4 ,! id5tities abv c 2 /at$ simply
0say+ t cos9e is an ev5 func;n1 & t s9e
is an odd func;n4

                                     #aa
  ,ev5 func;ns >e symmetric acr ! ;y-
axis2 ! left & "r sides >e id5tical4
,figure #12 %[s a typical ev5 func;n4
  ,odd func;ns >e symmetric ab ! orig94
,t is1 ! graph ( an odd func;n is
un*ang$ if y /ick a p9 9 ! orig9 &
rotate ! page #180 9 ! plane (! pap]4 ,a
typical odd func;n is %[n 9 ,figure #13
_4
  ,a la/ basic id5t;y cd 2 se5 0look+ at
! graphs (! s9e & cos9e func;ns 9
,figures #9 & #10_4 ,! two graphs h !
same %ape1 except t ! graph (! cos9e
func;n seems 6be ! graph (! s9e func;n
slid left 0a 4t.e .p_/2_4 ,we saw 9 ! f/
lab t 6slide a graph left 0a 4t.e ;k y
add ;k 6! value (! >gu;t4 ,?us1
  cos .? .k sin (.?+?.p_/2#)
  or1 equival5tly1
  sin .? .k cos (.?-?.p_/2#)_4
  ,6prove ^! id5tities m -pell+ly f !
unit circle1 ?9k ab ! picture 9 ,figure
#14, : %[s ! angle .?, ! angle
.?+?.p_/2#, &! po9ts ^! angles cross !
                                     #ab
unit circle4 ,x is cle> geometric,y t !
;y coord9ate ( po9t at : ! l>g] angle
meets ! unit circle is ! same z ! ;x
coord9ate (! po9t at : ! small] angle
meets ! unit circle1 i4e41 t
sin (.?+?.p_/2#) .k cos .?_4
  ,= -plete;s1 let me li/ #2 m id5tities
9 ! same family z ^!3
  -cos .? .k sin (.?-?.p_/2#),
  -sin .? .k cos (.?+?.p_/2#)_4
  ,^! >e easy 3sequ;es (! id5tities we
alr h4

,o!r ,trig ,func;ns
  ,9 a4i;n 6! s9e & cos9e1 "! >e #4 o!r
trig func;ns1 ! secant1 cosecant1
tang5t1 & cotang5t1 : c 2 def9$ z
  sec .? .k ?1_/cos .?#
  csc .? .k ?1_/sin .?#
  tan .? .k ?sin .?_/cos .?#
  cot .? .k ?cos .?_/sin .?#
  ,! graphs (! secant & cosecant func;ns
c easily 2 obta9$ f ! graphs ( s9e &
cos9e4 ,s9ce sec .? .k ?1_/cos .?#, we
                                     #ac
"k t :5 cos .? is sli<tly less ?an #1,
sec .? w 2 sli<tly m ?an #1_4 ,:5 cos .?
is small & positive1 sec .? w 2 l>ge &
positive4 ,simil>ly1 :5 cos .? is
sli<tly bi7] ?an -#1, sec .? w 2 sli<tly
less ?an -#1, & :5 cos .? is small &
negative1 sec .? w 2 l>ge & negative4
,graphs ( y .k cos .? & y .k sec .? >e
plott$ tgr 9 ,figure #15, & graphs (
y .k sin .? & y .k csc .? >e plott$ tgr
9 ,figure #16_4 ,9 ^! plots1 ! s9e &
cos9e func;ns >e drawn z da%$ l9es1 &!
secant & cosecant >e drawn z solid l9es4
  ,! func;ns sec .? & csc .? >e less
important ?an sin .? & cos .?, s9ce x is
easi] 6?9k ( physical ?+s t look l waves
?an x is 6?9k ( physical ?+s t %oot (f
to ,=, reappe> at -,=, zoom up t[>d ! ;y
axis only 6ru% back (f to -,=, & !n
rep1t ? ag & ag4
  ,6graph ! tang5t func;n1 & 6see xs
import.e1 x's wor?:ile 6g back 6! unit
circle4 ,! po9t 9 ,figure #17 on ! unit
circle at an angle .? f ! ;x axis has
                                     #ad
coord9ates y .k sin .?, x .k cos .?_4
,?us1
tan .? .k ?sin .?_/cos .?# .k ?y_/x# is
! slope (! l9e 2t ? po9t &! orig94
  ,x's n[ easy 6sket* a graph (
y .k tan .?_4 ,:5 .? .k #0, ! slope (!
l9e is #0, s tan #0 .k #0_4 ,z .? gr[s1
! slope 9cr1ses1 until z .? approa*es
?.p_/2# .k #90, ! slope approa*es ,=_4
,/>t+ f #0 & decr1s+1 ! slope 2comes m &
m negative1 approa*+ -,= z .? gets close
to -?.p_/2# .k -#90_4 ,if .? is sli<tly
l>g] ?an ?.?_/2#, !n ! slope (! l9e is
l>ge & negative4 ,x 9cr1ses to #0 :5
.? .k .p, & 3t9ues 69cr1se 6approa* ,= z
.? gets close to ?3.p_/2#_4 ,figure #18
%[s a sket* ( y .k tan .?_4
  ,at a f/ gl.e1 tan .? looks "s?+ l x^3
ne> ! orig91 b "! >e subtle di6];es4
y .k x^3 has a slope ( #0 at x .k #0,
:ile y .k tan .? has a slope ( #1_4
  ,x's n h>d 6play ! same game 6get a
graph (! cotang5t1 %[n 9 ,figure #19_4

                                     #ae
,p]iodic;y
  ,we've notic$ t sin .? & cos .? rep1t
af #2.p .k #360_4 ,! same c easily 2 se5
6hold = sec .? & csc .?_4 ,on ! o!r h&1
a bit ( ?"\ or a gl.e at ! plots %[s t
tan .? & cot .? rep1t af on .p .k #180_4
,algebraic,y1 = e 9teg] ;n,
  sin (.?+2.pn) .k sin .?
  cos (.?+2.pn) .k cos .?
  sec (.?+2.pn) .k sec .?
  cot (.?+2.pn) .k cos .?
  tan (.?+.pn) .k tan .?
  cot (.?+.pn) .k cot .?

,basic id5tities =! o!r trig func;ns
  ,x's easy 6see look+ at ! graphs t
sec .? & cot .? >e ev5 func;ns1 & t
csc .? & tan .? >e odd func;ns4
  sec (-.?) .k sec .?
  cot (-.?) .k cot .?
  tan (-.?) .k -tan .?
  csc (-.?) .k -csc .?
  ,"! is al a use;l id5t;y = secant &
tang5t t resem#s ! ,py?agor1n id5t;y1 &
                                     #af
t is obta9$ f x4
  sin^2 .?+cos^2 .? .k #1
  ?sin^2 .?_/cos^2 .?#+?cos^2 .?_/
    cos^2 .?# .k ?1_/cos^2 .?#
  tan^2 .?+1 .k sec^2 .?
  ,p]son,y1 ,i c n"e rememb] :e!r !
id5t;y is tan^2 .?-sec^2 .? .k #1 (wr;g)
or :e!r x's sec^2 .?-tan^2 .? .k #1 ("r
)_2 s ,i j rememb] h[ 6prove ! id5t;y :5
,i ne$ x4

,3nec;ns ) triangles
  ,:y >e "! exactly #6 trig func;ns8
,:at motivat$ ! odd def9i;ns (! secant &
tang5t & s on8 ,& :at does trigonometry
h 6d ) triangles8 (_8,gnomon0 m1ns 8
angle0 9 ,greek4) ,6answ] ? "q1 ?9k ab
an >bitr>y "r triangle4 ,? triangle c 2
relat$ 6! unit circle z %[n 9 ,figure
#20_4
  ,! #2 triangles 9 ? figure >e simil>4
,! small] triangle has $ges
cos .?, sin .?, & #1_4 ,! correspond+
sides (! l>ge triangle >e tradi;n,y
                                     #ag
call$ ! adjac5t & opposite sides1 &!
hypot5use4 ,?9k+ ab simil> triangles n[
gives u
  sin .? .k ?sin .?_/1# .k ?opp_/hyp#
  cos .? .k ?cos .?_/1# .k ?adj_/hyp#
  tan .? .k ?sin .?_/cos .?# .k ?opp
    _/adj#
  sec .? .k ?1_/cos .?# .k ?hyp_/adj#
  cot .? .k ?cos .?_/sin .?# .k ?adj
    _/opp#
  csc .? .k ?1_/sin .?# .k ?hyp_/opp#
  ,! trig func;ns >e "!=e ! #6 possi#
ways 6select #2 di6]5t sides (a "r
triangle & 6take _! ratio4

,sum1 ,d\# & ,half ,angle ,=mulas
  ,9 hi< s*ool1 y may h se5 scores (
id5tities 3nect+ ! trig func;ns4 ,two
important "os we'll make ref];e 6once or
twice >e ! sum =mulas1
  cos (.?+.f) .k cos .?cos .f-sin .?
    sin .f
  sin (.?+.f) .k cos .?sin .f+sin .?
    cos .f
                                     #ah
  ,if we set .f .k .?, !n ^! 2come ! d\#
angle =mulas1
  cos (2.?) .k cos^2 .?-sin^2 .?
  sin (2.?) .k #2sin .?cos .?
  ,! f/ (! d\# angle =mulas c 2 trans=m$
6give
  cos (2.?) .k cos^2 .?-(1-cos^2 .?)
    .k #2cos^2 .?-1,
: c 2 solv$ 6give
  cos^2 .? .k ?1+cos (2.?)_/2#_4
  ,alt]natively1 "o cd write
  cos (2.?) .k (1-sin^2 .?)-sin^2 .?
    .k #1-2sin^2 .?,
: solves 6give
  sin^2 .? .k ?1-cos (2.?)_/2#_4
  ,= -plete;s (not = any imag9a# te/)1
let me %[ y ": ! sum =mulas -e f4
,figure #21 is a draw+ (! unit circle1 )
#4 po9ts %[n4 ,po9t ;,x is ! po9t (1, 0)
at : ! ;x axis meets ! unit circle4
,po9t ;,b is an angle .? abv ;,x_4 ,po9t
;,a is an angle .f abv ;,b, & "!=e an
angle .?+.f abv ;,x_4 ,po9t ;,c is an
angle .f 2l ;,x, & "!=e an angle .?+.f
                                     #ai
2l po9t ;,b_4
  ,! coord9ates (! po9ts 9 ! ,figure >e
"!=e
  ,x .k (0, 0)
  ,b .k (cos .?, sin .?)
  ,a .k (cos (.?+.f), sin (.?+.f))
  ,c .k (cos (-.f), sin (-.f))
    .k (cos .f, -sin .f)
  ,n[1 ! po9ts ;,a & ;,x >e ! same 4t.e
a"p z ! po9ts ;,b & ;,c, s9ce !y >e
sep>at$ 0! same angles on ! unit circle4
,! ,py?agor1n ,!orem tells u t ^! 4t.es
>e
  ",a,x<:]^2
    .k @(cos (.?+.f)-1@)^2"+@(sin (.?
      +.f)-0@)^2
    .k cos^2 (.?+.f)-2cos (.?+.f)+1+
      sin^2 (.?+.f)
    .k #2-2cos (.?+.f)
&
  ",b,c<:]^2
    .k @(cos .?-cos .f@)^2"+@(sin .?+
      sin .f@)^2
    .k cos^2 .?-2cos .?cos .f+cos^2 .f+
                                     #bj
      sin^2 .?+2sin .?sin .f+sin^2 .f
    .k #2-2cos .?cos .f+2sin .?sin .f_4
,sett+ ^! #2 expres.ns equal gives
  -#2cos (.?+.f) .k -#2cos .?cos .f+2
    sin .?sin .f
  cos (.?+.f) .k cos .?cos .f-sin .?
    sin .f
: is ! f/ (! sum =mulas4 ,6get ! second
sum =mula1 use apply ! id5t;y
sin .a .k cos (?.p_/2#-.a) 6! f/ sum
=mula4 ,y get
  sin (.?+.f)
    .k cos (?.p_/2#-(.?+.f))
    .k cos (?.p_/2#-.?)cos (.f)-sin (?.p
      _/2#-.?)sin .f
    .k sin .?cos .f+sin .?cos .f_4

,special ,values
  ,=a few values ( .?, we c easily -pute
sin .? & cos .? exactly4 ,if
.? .k ?.p_/4# .k #45, !n ! po9t (x, y)
lies bo? on ! unit circle x^2"+y^2 .k #1
& on ! l9e y .k x_4 ,? m1ns t
x^2"+x^2 .k #1, s x^2 .k ?1_/2#, s
                                     #ba
x .k cos ?.p_/4# .k ?1_/>2]#
.k ?>2]_/2#_4 ,s9ce y .k x,
y .k sin ?.p_/4# .k ?1_/>2]# .k ?>2]_/2#
z well4
  ,if .? .k ?.p_/3# .k #60, !n x is n
h>d 6see geometric,y t "! is an
equilat]al triangle pres5t 9 \r usual
figure1 z %[n 9 ,figure #22_4
  ,x is cle> f ! figure t
x .k cos ?.p_/3# .k ?1_/2#_4 ,we c !n
solve x^2"+y^2 .k #1 (the equ,n (! unit
circle) 6get
y .k sin ?.p_/3# .k ?>3]_/2#_4 ,a simil>
3/ruc;n lets "o -pute ! s9e & cos9e (
?.p_/6# .k #30, or y c get !m f !
id5tities we've alr deriv$4 ,! results
>e summ>iz$ 9 ! ta# 2l4

      ,'2g9.(c5t].),'





                                     #bb

---------------------------------
.?               cos .?   sin .?
---------------------------------
#0               #1       #0
---------------------------------
.p_/6 .k #30     >3]_/2   #1_/2
---------------------------------
.p_/4 .k #45     #1_/>2]  #1_/>2]
---------------------------------
.p_/3 .k #60     #1_/2    >3]_/2
---------------------------------
.p_/2 .k #90     #0       #1
---------------------------------
#2.p_/3 .k #120  -#1_/2   >3]_/2
---------------------------------

      ,'5d.(c5t].),'






                                     #bc