1. trigonometry (ft)
TRANSCRIPT
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Core 2
Trigonometry
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Lesson Objectives
Convert between radians and degrees Find the length of an arc and the area of a
sector Find the area of a triangle and a segment Use the sine and cosine rules.
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DEGREESare not the only way to measure angles.
A RADIANis a larger unit which is often used in
trigonometry because it can simplify many
calculations.
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Consider a circle with radius 1.
The angle POQ in radians is
the same as the distance you
would travel from P to Q along
the circle.
1
1
O
Q
P
egrees !adians
360o
10o
!0o
"#o
30o
2
/2
/4
/6
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"ow to convert radians into degrees and vice#versa$
egrees !adians
egrees !adians
180
180
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$%am&le
Convert to radians
o
30
o
120o
60
o
45
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$%am&le
Convert to radians
o
72
o
720o
315
o
36
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'ractice
(egrees and radians dominoes
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Lesson Objectives
Convert between radians and degrees Find the length of an arc and the area of a
sector Find the area of a triangle and a segment Use the sine and cosine rules.
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%f the radius doubles& the arc
length also doubles& so we canuse a simple formula to
calculate the arc length
The length of an arc
rO
Q
P
%f the circle has radius 1& thearc length is the same as the
angle .
Arc PQ ' r
"owever& if we increase or
decrease the radius& the arclength will change& but willremain the same.
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EXAMPLE 1
Calculate the arc length and perieter o! a sector o!
angle 2()and radius " c# Lea$e %our answers inters o! #
2()* *)rc length * r
)rc length * 6 X 2
()
)rc length * "
'erimeter * "+ , 6 , 6 * "+ , 1-
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Area of a sector
+e only want the sector
with angle
The area of the circle is ' r2
Area of sector ' 1(2r2
r
Q
O
PThis would be a sector with
angle 2
,o we can use the formula$
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EXAMPLE 1
Calculate the area o! a sector o! angle 2()and radius
" c# Lea$e %our answers in ters o! #
2()* *
Area of sector ' 1(2r2
Area of sector ' 1(2X *2 X 2()
Area of sector ' 12
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$%ercise
Form the cards into a loo& in our grou&s.
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Lesson Objectives
Convert between radians and degrees Find the length of an arc and the area of a
sector Find the area of a triangle and a segment Use the sine and cosine rules.
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)rea of a triangle
/his also wors if we measure in radians butwe need to change the mode of our calculator2
hift
4odeO&tion for 5ad "7
/o change bac to degrees
hift4odeO&tion for (eg 37
Cabsin21
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)rea of a triangle
e.g. Find the area of this triangle to 3significant figures
-/3 ! cm6 cm
)rea * 8 absinC)rea * 8 6 X !X sin-/37
)rea * -3."cm-
Cabsin21
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EXAMPLE 1
&or this circle' centre (' radius ) c and
angle A(* +,
(-!ind the area o! the inorsegent to - d#p#
o
A
##
Area of sector ' 1(2r2
Area of sector '1
(2/2
X,
(-Area of sector ' .)(0
)rea of triangle * 8 absinC
)rea of triangle * 8 #X#Xsin,(-7)rea of triangle * .33"96#
)rea of segment * area of sector : area of
triangle)rea of segment * -0.61cm-
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5eminder
Convert 1#0ointo radians
Convert into degrees
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$%ercise
Form the cards into a loo& in our grou&s.
1# minutes
EXAMPLE - /wor0 in groups
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P 18 yds12 yds
10 yds
EXAMPLE - /wor0 in groups
2he area on a !oot3all pitch 0nown as the 4D5
is 3ounded 3% the 167%ard line and the arc o! a
circle' radius 18 %ards with its centre at thepenalt% spot /P' as shown in the diagra# 2he
penalt% spot is 1- %ards !ro the goal line#
Calculate the area o! the 4D5 to -d#p#
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12 yds
18 yds
10 yds10 yds
6 yds
8yds8 yds
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6 10
Cos=6/10
=cos-1(6/10)
=cos-1(6/10)
=0.92729528
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10ds10ds
16ds
1.#c
)rea of sectorector )rea * 8 r-;
ector )rea * 8%10-%1.#ector )rea * !-.#ds-
)rea of triangle
/riangle )rea * 8absinC/riangle )rea * 810%10sin1.#
/riangle )rea * ".06ds-
)rea of ()rea of ( * ector )rea
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Lesson Objectives
Convert between radians and degrees Find the length of an arc and the area of a
sector Find the area of a triangle and a segment Use the sine and cosine rules.
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ine rule and cosine rule
=oth these rules wor in degrees andradians
C
c
B
b
A
a
SineRulesinsinsin
: ==
booklet)formulae(in
cos2:sin 222 AbccbaeRuleCo +=
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Lesson Objectives
Convert between radians and degrees Find the length of an arc and the area of a
sector Find the area of a triangle and a segment Use the sine and cosine rules.
>oring in radians
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>oring in radians
Find angle C Use the sine rule
)
=
C
1
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Trigonometry -Gra!s?ra&hs of /rig. Functions
* sin % * cos % * tan %
Using trig gra&hs to solvee@uations
http://var/www/apps/conversion/tmp/scratch_4/y=sinx.agghttp://var/www/apps/conversion/tmp/scratch_4/y=cosx.agghttp://var/www/apps/conversion/tmp/scratch_4/y=tanx.agghttp://var/www/apps/conversion/tmp/scratch_4/y=tanx.agghttp://var/www/apps/conversion/tmp/scratch_4/y=cosx.agghttp://var/www/apps/conversion/tmp/scratch_4/y=sinx.agg -
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Gra!s o" Trig #$nctions
*sin%
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Gra!s o" Trig #$nctions
*cos%
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Gra!s o" Trig #$nctions
*tan%
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%o&'ing $ations
1. *inear e$ations+ %o&'e , 2 = 0
1 so&$tion
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%o&'ing $ations
2. $aratic e$ations+ %o&'e ,22 = 0
1 so&$tion2 so&$tions
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%o&'ing $ations
,. Trig e$ations+ %o&'e %in = 0.5
many so&$tions
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%o&'ing Trig $ations $sing
Gra!s1. %o&'e %in = 0.3 "or ,604 ,604
i. se yo$r ca&c$&ator to "in t!e
:;*
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'rinci&al value *-3.#A
2,.58-,60=-,,6.32
156.32-,60=-20,.58
180-2,.58=156.32
For sine 10 : &rinci&al value
ine re&eats ever 360o se we can add or subtract360 from our answers to find more solutions in the
given interval
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2. %o&'e :os = -0.5 "or ,604 ,604
,. %o&'e Tan = 1.2 "or ,604 ,604
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'rinci&al value *1-0A
120-,60=-230
230-,60=-120,60-120=230
For cosine 360 : &rinci&al value
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'rinci&al value* #0.1!A
50.19-,60=-,09.8150.19-180=-129.81
50.19180=2,0.19
For tan 10 , &rinci&al value
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o> try t!e "o&&o>ing+
%o&'e t!e "o&&o>ing e$ations "or04 ,604gi'ing yo$r ans>er to t!e nearest 4+
a. tan = 1?. cos = 0.5c. tan = -1. cos = -0.9e. sin = -0.25". cos = -1
354@ 2254@604@ ,004@
1,54@ ,154
1534@ 2064
1934@ ,364
1804
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aians
Ae !a'e a&reay seen t!at >e canmeas$re ang&es in raians@ so >e canso&'e trig e$ations $sing raians.
T!e gra!s &oo t!e same.
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*sin%
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*cos%
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*tan%
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olving e@uations in radians
/o change the setting on ourcalculator &ress
hift 4ode " 5ad or 3 (eg
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%o&'ing Trig $ations $sing
Gra!s1. %o&'e %in = 0.3 "or 2B 2Bi. se yo$r ca&c$&ator to "in t!e
:;*
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'rinci&al value *0."1-
0.312-2C =-5.87
2.7,-2C=-,.55
C-0.312=2.7,
For sine + : &rinci&al value
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2. %o&'e :os = -0.5 "or 2B 2B
,. %o&'e Tan = 1.2 "or 2B 2B
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'rinci&al value *-+B3
-+B3--+=-3+B3
"+B3--+=--+B3 -+
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'rinci&al value* 0.96
0.96 < -+ *-5.307 0.96 : +*
-2.266
0.96,+ *3.018
For tan + , &rinci&al value
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o> try t!e "o&&o>ing+
%o&'e t!e "o&&o>ing e$ations "or04 2Bgi'ing yo$r ans>er to , s.".
a. tan = 1.5?. cos = 0.6c. tan = -0.8
. cos = -0.,e. sin = -0.35". sin = 0.7
0.98,@ 3.120.927@ 5.,6
2.37@ 5.61
1.88@ 3.31
,.61@ 5.82
0.775@ 2.,7
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%D*
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). Quadratic trig euations$
$%am&les1. olve sin-; * D for 0E ; 360E.-. olve cos-; * 8 for 0E ; 360E.3. olve tan-% : tan % * 0 for
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). Quadratic trig euations$
$%am&lesolve sin-; * D for 0E ; 360E.@uare root sin; * G8
olve each e@uation se&aratelsin; * 8 sin; *
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). Quadratic trig euations$
$%am&lesolve cos-; * 8 for 0E ; 360E.@uare root cos; * GH8
olve each e@uation se&aratelcos; * H8 cos; * < H8; * cos
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). Quadratic trig euations$
$%am&lesolve tan-% : tan % * 0 for
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). Quadratic trig euations$
$%am&lesolve -cos-% , #cos % , - * 0 for
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y%o&'e t!e "o&&o>ing e$ations in t!e
inter'a& 0F ,60F.a. 6sin2 sin -1 = 0
?. 3cos2 7cos = 2
c. 6cos2 cos 1 = 0
. 3sin2 ,sin = 1
,0@[email protected]@,30.5
75.5@ 283.5
70.5@ 120@[email protected]
13.5@ 165.5@ 270
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). ,in a& cos a and tan a
$%am&lesolve cos -% * 8 for 0E % 360E.(ouble the domain 0E -% 9-0E
olve as normal then divide b - at the endcos -% * 8 -% * cos
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). ,in a& cos a and tan a$%am&lesolve "sin -% , 3 * 0 for
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). ,in a& cos a and tan a$%am&lesolve tan3%*1 for :+ J % J +/reble the domain :3+ J 3% J 3+
olve as normal then divide b 3 at the endtan 3% * 1 3% * tan
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=. sin3 4 a5& cos3 4 a5& tan3 4 a5$%am&le
olve sin % < 60E7 * 0." for
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$%am&le
olve * 0.# for
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$%am&le
olve tan -% < 30E7 * 1 for
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$%ercise
4atch the cards together in our grou&s
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/rue for)LLvalues of %
TG HTT%+
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Kdentit 1
/here is an identit that lins sin% cos%and tan% together.
ou do not need to now the &roof butou doneed to now the identit2
i
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sin%*o&& h&
cos%*adj h&
tan%*o&& adj
=ut what is sin% cos%
h&otenuse
adjacent
o&&osite
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x
x
x
cos
sintan
T i f f P h
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Trig form of Pythagoras
)&&ling 'thagorasto the triangle gives
I
I
1
,in 6
Cos 6sin2 + cos2 = 1
1. ?iven that
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a. find the value of
b. hence solve the e@uation for
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Use in the form
Io solutions2
3. how that the e@uationCan be written as
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Can be written as
Mence show that this can be rewritten as
Mence solve the e@uation for 0J%J360E.
". olve the e@uation for
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Use
$ i
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$%ercise
/ae a card Cover u& the answer and hints /r the @uestion if ou get stuc loo at
the hints and then chec our answer. 'ut the card bac in the &ile and tae
another.
Guess Who!
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Guess Who!
You have 7 minutes tofind the answers for 0
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cos = 0.5 Tan2= 1 sin(,0)=0.3
6.sin(,-,0)=0.,
7.tan(75) =,
8.3cos2 7cos = 2
9.cos = 2
10.
11.cos2 = 1
12.tan =3
1,.
cos(2-10)=0.7 13.tan 2 = 0 15.,sin=7cos
16.
3sin,=0
17.6cos2
cos 1 = 0
18.
sincos=0
19.
sin(tan3)=0
20.(sin-,)(tan,)
=0
21.sin(5-25)=8
22.8cos=-,sin
2,.,-5cos=0
23.3sin2 ,sin = 1
25.8sin,=1
-1 = 0 sin = ,
,.3.
22 5 112 55.
1.,0@[email protected]@ 2.
&
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60@ ,[email protected]@
[email protected]@ ,5,.6
6.
sin(,-,0)=0.,15.8@1,[email protected]@183@,03
7.tan(75) =,
177@,57
8.3cos2 7cos = 275.5@ 283.5
9.cos = 2
o so&$tions
10.
31.8@ 1,8
11.cos2 = 1
180
12.Tan =376.0@ 256
1,.cos(2-10)=0.7
27.8@208@162@,32
13.tan 2 = 06,.3@ 23,
15.,sin=7cos
66.8@237
16.
3sin,=0229@,11
17.
70.5@ 120@[email protected]
18.
sincos=01,5@,15
19.sin(tan3)=
0180@103@283
20.(sin-,)(tan,)
=0108@288
21.sin(5 -25)=8
o so&$tions
22.8cos=-,sin
111@291
2,.,-5cos=0
5,.1@,07
23.13.5@ 165.5@
270
,30.5 o %o&$tions
25.8sin,=1
,0@150
$ l
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$%am&les
1. Find the value of tan ; when sin ; * 3B# andcos ; *
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