1. Trigonometry (FT)

7/24/2019 1. Trigonometry (FT)

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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




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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(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




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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

(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




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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



>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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*cos%


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*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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$%am&lesolve sin-; * D for 0E ; 360E.@uare root sin; * G8

olve each e@uation se&aratelsin; * 8 sin; *


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$%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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$%am&lesolve tan-% : tan % * 0 for


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$%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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1. Trigonometry (FT)

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