X
Class
Introduction to Trigonometry
Chapter
1. Acquires Acquires the knowledge knowledge of terms like angle ,trigonometry ,trigonometry ,
%earning &$'ecti!e
Sine, cosine and tangent of angle related with trigonometry. trigonometr y. . To de!elop de!elop the understanding understanding of the processes and and principles underlying underlying the formation of trigonometric ratios of gi!en angle. ". To de!elop de!elop the understanding understanding of the processes and and principles underlying underlying trigonometric ratios of complementary angles and trigonometric identities. identities. # To apply the knowledge of trigonometric ratios for sol!ing different types of pro$lems.
Students should ha!e the knowledge of the concept of ratio and know that )re!ious di!isi di!ision on is not not commu commutat tati!e i!e.. Studen Students ts should should ha!e ha!e studie studiedd ()yth ()ythago agoras ras -nowledge Theorem* and know the meaning of the term +hypotenuse. Trigonometric atios
Topic
Trigonometric ratios of some specific angles Trigonometric ratios of complementary angles Trigonometric identities
/ou ha!e already studied a$out triangles, and in particular right triangles, in your earlier classes. %et us take some e0amples from our surroundings where right triangles can $e imagined to $e formed. Suppose, if you are looking at the top of an electrical pole, a right triangle can $e imagined
Introductio n
$etween yours*s eye and the top of the pole Can you find out the height of the pole, without actually measuring it
pole o$ser!er 2i!e some e0amples for o$'ects you find around you which are in the shape of a right angled triangle In all the situations gi!en a$o!e, the distances or heights can $e found $y using some mathematical techniques, which come under a $ranch of mathematics called (trigonometry*. The word (trigonometry* is deri!ed from the 2reek words (tri* 3meaning three4, (gon* 3meaning sides4 and (metron* 3meaning measure4. In fact, trigonometry is the study of relationships relationships $etween the sides and angles of a triangle. The earliest known work on trigonometry was recorded in 5gypt and 6a$ylon. 5arly astronomers used it to find out the distances of the stars and planets from the 5arth. 5!en today, most of the technologically ad!anced methods used in 5ngineering and )hysical Sciences are $ased on trigonometrical concepts. In this chapter, we will study ratios of the sides of a right triangle with Content respect to its acute angles, called trigonometric ratios of the angle. 7e will restrict our discussion to acute angles only. 8owe!er, these ratios can $e e0tended to other angles also. 7e will also define the trigonometric ratios for angles of measure 9: and ;9:. 7e will calculate trigonometric ratios for some specific angles and esta$lish some identities in!ol!ing in!ol!ing these ratios, called trigonometric identities. The three sides of a right triangle are called 1. )erp )erpen endi dicu cula larr, . 6ase 3 the the side side on which which perpen perpendicu dicular lar stand stands4 s4 ". 8ypotenu 8ypotenuse se 3the side side opposite opposite to to the right right angle4. angle4.
Since sum of angles in a triangle is 1<9 9 and one angle is ;99, the other two angles ha!e to $e necessarily acute 3=;9 94angles. The acute angles 3two in num$er4 are normally denoted $y 2reek letters alpha 3 4, $eta 3 4, gamma 3 4, theta 3 In the ad'oining figure X)/ is a right angles triangle with X)/ > ;99 7e also notice that SA/ T6/ ? @C/ X)/. Thus $y similarity similari ty property of SA/ and T6/, /A/6 >/S/T>AS6T >/S/T>AS 6T /A/S>/6/T> 6ase8ypotenuse /AAS>/66T> 6ase)erpendicular AS/S>6T/T> )erpendicular 8ypotenuse
Since these ratios are constant irrespecti!e of length of the sides we represent represent these ratios $y some standard names. Since the right triangle has three sides we can ha!e si0 different ratios of their sides as gi!en in the following ta$leB
e
Short form
atio of sides
/
sin /
)erpendicular 8ypotenuse 6ase8 ase8ypotenu se )erpendicular 6ase 8ypotenuse)e rpendicular 8ypotenuse$a se 6aseperpendic ular
e/
cos /
ent /
tan /
cant / nt /
cosec / sec /
ngent
cot /
In the Digur e >)X /X >/) /X >)X /) >/X )X >/X /) >/) )X
emarks 3)84 3684 >sin / cos /,3)64 /,3)64 >1sin / >1cos / >1tan/>cos/sin/
hree ratios 3#, E and F4 are reciprocals reciprocals 3in!erse4 of t three ratios, GCosecA >1 GSecA >1 CotA >1 ng 3Identification4 of $ase and perpendicular sides are angea$le depending upon upon the angle angle opposite to the the spect to X, the $ase is X) and perpendicular perpendicula r is )/. spect to /, the $ase is /) and perpendicular perpendicul ar is )X4. lso called (opposite side*of side* of / and /) is led (ad'acent side* of /4
you need to memoriHe this simple mnemonic Some people have have curly brown hair hair turned permanently black black
That*s all you need to memoriHe to register the trigonometrical ratios in your mind fore!er. So here you go, Some )eople 8a!e S > )8 Sin > )erpendicular 8ypotenuse Curly 6rown 8air C > 68 Cos > 6ase8ypotenu 6ase8ypotenuse se Turned )ermanently 6lack T > )6 Tan > )erpendicular6ase There are " more ratiosB Cosec, Sec and Cot. Dor these, 'ust remem$er that Cosec is the reciprocal of SinJ or Cosec > 1Sin > 8) Sec is the reciprocal of CosJ or Sec > 1Cos > 86 Cot is the reciprocal of TanJ or Cot> 1Tan > 6)
.
The standard angles of trigonometrical ratios are 9:, "9:, #E:, F9: and ;9:. The !alues of trigonometrical ratios of standard angles are !ery important i mportant to sol!e the trigonometrical trigonometrical pro$lems. Therefore, Therefore, it is necessary to remem$er the !alue of the trigonometrical ratios of these standard angles. The sine, cosine and tangent of the standard angles are gi!en $elow in the ta$le
. To remem$er the a$o!e !aluesB 3a4 di!ide the num$ers 9, 1, , " and # $y #, 3$4 take the positi!e square roots, 3c4 these num$ers gi!en the !alues of sin 9:, sin "9:, sin #E:, sin F9: and sin ;9: respecti!ely. 3d4 write the !alues of sin 9:, sin "9:, sin #E:, sin F9: and sin ;9: in re!erse
order and get the !alues of cos 9:, cos "9:, cos #E:, cos F9: and cos ;9: respecti!ely. If K $e an acute angle, the !alues of sin K and cos K lies $etween 9 and 1 3$oth inclusi!e4. The sine of the standard angles 9:, "9:, #E:, F9: and ;9: are respecti!ely the positi!e square roots of 9#,1#, #,"# and ## Therefore, sin 9: > L39#4 > 9 sin "9: > L31#4 > M sin #E: > L3#4 > 1L > L sin F9: > L"# > L"J Cos ;9: > L 3##4 > 1. Similarly cosine of the above standard angels are respectively the positive square roots of 4/4, 3/4, 2/4, 1/4, 0/4
Complementary angles in rigonometry Complementary angles in trigonometryB Two angles are said to $e complementary, if their sum is ;9 9. It follows from the a$o!e definition that K and 3 ;9 K 4 are complementary angles in trigonometry for an acute angle K In NA6C, ∠6 > ;9 9 9 ∴ ∠A O ∠C > ;9 9 ∠C >;9 ∠A
Dor the sake of easiness in this deri!ation, we will write ∠C and ∠A as C and A respecti!ely Thus C > ;9 9 A sin A > 6C AC cosec A > AC 6C cos A > A6 AC sec A > AC A6 tan A > 6C A6 cot A > A6 6C and sin C > sin 3;9 9 A 4 > A6 ACJ cosec C > cosec 3;9 9 A4 > AC A6 cos C > cos 3;9 9 A4 > 6C ACJ sec C > sec 3;9 9 A4 > AC 6C tan C > tan 3;9 9 A4 > A6 6CJ cot C > cot 3;9 9 A4 > 6C A6 sin 3;99 A4 > cos A tan3;9 9 A4 > cot A sec3;99 A4 > cosec A cos 3;99 A4 > sin A cot3;9 9 A4 > tan A cosec 3;99 A4 > sec A Some Sol!ed 50amples on complementary angles in trigonometry B 14 5!aluate B cos "P 9 sin E" 9 Solution B cos "P 9 sin E" 9 > cos3 ;9 Q E" 4sin E" > sin E" 9 sin E" 9 > 1 RRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRR RRRRRR 4 Show that B 3 cos P9 9 4 3sin 9 9 4 O 3cos E; 9 4 sin "1 9 < sin "9 9 > 9 Solution B Consider
3 cos P9 9 4 3sin 9 9 4 O 3cos E; 9 4 sin "1 9 < sin "9 9 > cos 3 ;9 9 9 9 4 sin 9 9 O cos3;9 9 "1 9 4 sin "1 9 < 0314 > sin 9 9 sin 9 9 O sin "1 9 sin "1 9 < 0 1 # >1O1Q >9
)roof of the Trigonom rigonometri etricc ide identit ntities ies )roof. According to the )ythagorean theorem,
x O y > r .
. . . . . . . . . . . . . . .314
Therefore, Therefore, on di!iding $oth sides $y r , x r
2
2
O
y r
2
2
>1
That is, according to the definitions definitions,, cosθ O sinθ > 1. . . . . . . . . . . . . .34 Apart from the order of the terms, this is the first trigonometric &n di!iding line 34 $y cos θ , we ha!e
That is, 1 O tanθ > secθ . And if we di!ide $y sin θ , we ha!e
That is, 1 O cotθ > cosecθ . .
&$'ecti!e B Dinding trigonometric trigonometric ratios ratios for "9 9,#E9 and F9 9 FA
Uaterial equiredB )astel sheet, geometry $o0
ACTIVIT
)rocedureB A 1. Vraw an isosceles right triangle on a colorful paper and paste it on pastel sheet as shown
. Calculate the length of hypotenuse using )ythagoras theorem. ".
∠
6CA >
∠
6AC > #E9
#. @sing the triangle determine the !alue of Sin#E 9, Cos#E9 and Tan#E 9 6. 1. Vraw an equilateral triangle triangle on a colorful paper and paste it on pastel sheet as shown
. Vraw )T perpendicular to W ". WT > T > cm #.Dind )T using )ythagoras theorem E. Calculate F. @sing
∆
∠
T)W
)TW, determine the !alue for
SinF99, Cos F99 and Tan F9 9 Vetermine the Sin"99, Cos "99 and Tan "9 9 using the same triangle &$ser!ationB Sin#E9 >1L , Cos#E9 >1L and Tan#E Tan#E 9 > 1 SinF99 >L", CosF99 >1 and TanF9 TanF9 9 > L" Sin"99 >1 , Cos"99 >L"and Tan"9 Tan"9 9 >1 L"
u$rics for ecording acti!ity work 31E marks4 )arameters
Uarks allotted
6ring material for acti!ity
1
Takes interest in class
1
Is Is regular
1
%isten , o$ser!es attenti!ely
1
Takes care of property
1
G)erformance of Acti!ity
19
Complete and Correct task
eeds help Independe to complete ntly works task $ut incomplete task
Tries to ust $egin make the task effort $ut incomplete task
19
<
E
F
"
1. 2i!e some e0amples for o$'ects you find around you which are in the shape of a right angled triangle.
!ecapitulati on
. 7hy is it necessary to ha!e only right angled triangle as the $asis for computing trigonometric ratios ".8ow are the three fundamental ratios sine, cosine and tangent defined In a right angled triangle Assignments %5Y5% 1 1. If triangle triangle A6C A6C is right right angled angled at 6 and A6 > 1 cm , 6C 6C > E cm ,
Dind 3I4 Sin A , Cot A. 3ii4 Cosec C, Cos C. 3
. If co cot A>
, find all other trigonometric ratios of the angle A.
4
sin A + cos A
". If E cos cos A Q 1 sin sin A > 9 ,find ,find the the !alue !alue of #. Simp Simpli lify fy B 3 1 O tan tan
θ
4 31 Q sin
θ
COSA – sin A 2 COSA θ
4 31 O sin
4
E. 5!al 5!alua uate teBB SinF SinF99 9 . Cos"99 O Sin "99 . Cos F99 F. If A > F99 and 6 > "9 9 !erify that Cos 3A64 3A64 > CosA Cos6 O SinA Sin6 P. )ro! )ro!ee that that B 3 1O Cot Cot
θ
43 1 Q Cos
θ
431 O Cos
θ
4 >1
<. If Sin 3A O 64 > 1 and Cos Cos 3A Q64 >1 Dind Dind A and 6. 5
;. 2i!e 2i!enn tanA tanA> >
4
, 399Z A Z ;9 94, find the !alue of 1 sin A cos A.
19.If sec A > , find the !alue of cot A Q 1.
%5Y5% sin A + cos A
1. If E cos cos A Q 1 sin sin A > 9 ,find ,find the the !alue !alue of . If tan A >
". If cot
∅
√ 2
1 , show that
3
>
4
tan A
, pro!e that
1 + tan
√
cosec
2
2
sec
A
COSA – sin A 2 COSA
>
√ 2 4
2
∅− cot ∅ 2
∅− 1
>
√ 7 3
#. If tan A > M, then pro!e pro!e that that M sin A cos A > 1E
E. If cotA cotA >#" >#"
c heck heck
1 Q tanA > cotA Q sinA
1 O tanA F. An equilater equilateral al triangle triangle is inscri$ inscri$ed ed in a circle circle of radius radius F cm. Dind its side. P. Din Dindd the the !alue !alue of of Ta Tan F99 geometrically. θ
<. If Co Cos
θ
Sin
θ
> 1, Show that Cos
O Sin
θ
>1 or 1
;. )ro! )ro!ee that that Tan 19 Tan 9 Tan "9[[. Tan <;9 > 1 19 )ro! )ro!ee that that 3 Sin Sin A O Sec Sec A4 A4 O 3CosA O CosecA4 > 3 1 O SecA . CosecA4. %5Y5% " 1. If Tan 3A O 64 > 1 and Sin 3A Q 64 >1, Dind A and 6. θ
. If Cos
Sin
θ
> 0 and Cos
θ
" Sin
θ
> y pro!e that
0 O y Q 0y > E ". If Sec
θ
O Tan
θ
θ
> p, pro!e that Sin
>
p p
2
2
+1
−1
#.Compare the area of the right angled triangles A6C and V5D in which ∠
A > "99 ,
∠
6 >;99, AC >#cm,
∠
V > F99 ,
∠
V5> #cm E.If tan K O sin K > m and tan K sin K > n ,show that 3m Q n 4 > 1F mn 1
F. )ro!e that 3cosec K Q sinK4 3sec K Q cosK4 >
tan θ + cotθ
P. )ro!e that sin K 31O tanK4 O cos K31OcotK4> K31OcotK4> sec K OcosecK <. If P sin K O " cos K > # find the !alue of sec K OcosecK
5 >;99 and
;. )ro!e that sec K sec# K cosec K O cosec# K > cot# K tan# K 19. )ro!e that SinFA O CosFA > 1 " Sin A Cos A
)&SSI6%5 5&S and their 5U5VIATI&S 1 7riting Sin ;9
θ
instead of Sin 3;9
θ
emediation
4 It has to $e emphasiHed that
for all complementary angles it has to $e written Sin 3;9
θ
4 with in
$rackets. Some students draw other types of triangles instead of right angled triangles for calculating trigonometric ratios Instruct the students that only right angled triangles are to $e used and hypotenuse is related to right triangles only. " Some students do not mention the angle along with the trigonometric ratio. sin
50
cos
> tan 5mphasiHe 5mphasiHe on writing ratio with correct correct angles angles and gi!e
sufficient practice. # Students consider Sin KO cos K > 1 and take its square root as Sin K O cos K > \1 Instruct the students that square root root is taken only for whole term i.e. not with O and Q sign E @se of trigonometric ratios to pro!e geometrical results, is not !ery common with students while this method $ecomes !ery useful in some of the
questions Students should $e encouraged to use trigonometric results in geometry, especially especially where the ratio of sides is gi!en.
To o$ser!e o$ser!e a relation $etween sum of squares of sine and cosine ratios of an 8ands on angle in a right angled triangle. acti!ity U5T8&VB 1.Consider three right angled triangles as shown
. ecords the results in the ta$le $elow
∆ ABC
Sin C >
Cos C >
Sin C O Cos C>
]V5D
Sin D >
Cos D >
Sin D O Cos D >
] )W
Sin >
Cos >
Sin O Cos >
&$ser!ationB Sin
θ
O Cos
θ
>1
Similarly we can show that 1 O tanθ > Sec
θ . And 1 O Cot θ >
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