lesson plan trigo class x by manisha

7/25/2019 Lesson Plan Trigo class x by manisha

1/20

X Class

Introduction to Trigonometry Chapter

1. Acquires the knowledge of terms like angle ,trigonometry ,

Sine, cosine and tangent of angle related with trigonometry.

. To de!elop the understanding of the processes and principles underlying

the formation of trigonometric ratios of gi!en angle.

". To de!elop the understanding of the processes and principles underlying

trigonometric ratios of complementary angles and trigonometric identities.

# To apply the knowledge of trigonometric ratios for sol!ing different types

of pro$lems.

%earning

&$'ecti!e

Students should ha!e the knowledge of the concept of ratio and know that

di!ision is not commutati!e. Students should ha!e studied ()ythagorasTheorem* and know the meaning of the term +hypotenuse.

)re!ious

-nowledge

Trigonometric atios

Trigonometric ratios of some specific angles

Trigonometric ratios of complementary angles

Trigonometric identities

Topic

/ou ha!e already studied a$out triangles, and in particular right triangles,in your earlier classes. %et us take some e0amples from our surroundingswhere right triangles can $e imagined to $e formed. Suppose, if you arelooking at the top of an electrical pole, a right triangle can $e imagined

Introductio

n


2/20

$etween yours*s eye and the top of the pole Can you find out the height ofthe pole, without actually measuring it

pole

o$ser!er2i!e some e0amples for o$'ects you find around you which are in the shapeof a right angled triangleIn all the situations gi!en a$o!e, the distances or heights can $e found $yusing some mathematical techniques, which come under a $ranch ofmathematics called (trigonometry*. The word (trigonometry* is deri!ed fromthe 2reek words (tri* 3meaning three4, (gon* 3meaning sides4 and (metron*3meaning measure4. In fact, trigonometry is the study of relationships

$etween the sides and angles of a triangle. The earliest known work ontrigonometry was recorded in 5gypt and 6a$ylon. 5arly astronomers used itto 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 withrespect to its acute angles, called trigonometric ratios of the angle. 7e willrestrict our discussion to acute angles only. 8owe!er, these ratios can $ee0tended to other angles also. 7e will also define the trigonometric ratios for

angles of measure 9: and ;9:. 7e will calculate trigonometric ratios forsome specific angles and esta$lish some identities in!ol!ing these ratios,called trigonometric identities.

The three sides of a right triangle are called1. )erpendicular,. 6ase 3 the side on which perpendicular stands4". 8ypotenuse 3the side opposite to the right angle4.

Content


3/20

Since sum of angles in a triangle is 1


4/20

Since these ratios are constant irrespecti!e of length of the sides werepresent these ratios $y some standard names.

Since the right triangle has three sides we can ha!e si0 different ratiosof their sides as gi!en in the following ta$leB


5/20

Shortform

atio of sides In theDigure

emarks

sin / )erpendicular

8ypotenuse

>)X

/X

3)84

/ cos / 6ase8ypotenuse

>/)/X

3684

/ tan / )erpendicular6ase

>)X/)

>sin / cos /,3)64

t / cosec/

8ypotenuse)erpendicular

>/X)X

>1sin /

/ sec / 8ypotenuse$ase

>/X/)

>1cos /

nt cot / 6aseperpendicular >/))X >1tan/>cos/sin/

e ratios 3#, E and F4 are reciprocals 3in!erse4 ofree ratios,

secA >1cA >1tA >1

3Identification4 of $ase and perpendicular sides aregea$le depending upon the angle opposite to the

ect to X, the $ase is X) and perpendicular is )/.ct to /, the $ase is /) and perpendicular is )X4.called (opposite side*of / and /) is(ad'acent side*of /4


6/20

you need to memoriHe this simple mnemonic

Some people have curly brown hair turned permanently black

That*s all you need to memoriHe to register the trigonometrical ratios inyour mind fore!er. So here you go,

Some )eople 8a!e

S > )8

Sin > )erpendicular 8ypotenuse

Curly 6rown 8air

C > 68

Cos > 6ase8ypotenuse

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)

.


7/20

The standard angles of trigonometricalratios are 9:, "9:, #E:, F9: and ;9:.

The !alues of trigonometrical ratios of standard angles are !ery importantto sol!e the trigonometrical pro$lems. Therefore, it is necessary toremem$er the !alue of the trigonometrical ratios of these standard angles.The sine, cosine and tangent of the standard angles are gi!en $elow in theta$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


8/20

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 $etween 9 and 1 3$othinclusi!e4.

The sine of the standard angles 9:, "9:, #E:, F9: and ;9: are respecti!ely thepositi!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 $ecomplementary, if their sum is ;9 9.It follows from the a$o!e definition that K and 3 ;9 K 4 are complementaryangles in trigonometry for an acute angle KIn NA6C, 6 > ;9 9A O C > ;9 9

C >;9 9 A


9/20

Dor the sake of easiness in this deri!ation, we will write C and A as Cand A respecti!elyThusC > ;9 9 Asin A > 6C AC cosec A > AC 6Ccos A > A6 AC sec A > AC A6tan A > 6C A6 cot A > A6 6Candsin C > sin 3;9 9 A 4 > A6 ACJ cosec C > cosec 3;9 9 A4 > AC A6cos C > cos 3;9 9 A4 > 6C ACJ sec C > sec 3;9 9 A4 > AC 6C

tan C > tan 3;99

A4 > A6 6CJ cot C > cot 3;99

A4 > 6C A6sin 3;99 A4 > cos Atan3;99 A4 > cot Asec3;99 A4 > cosec Acos 3;99 A4 > sin Acot3;99 A4 > tan Acosec 3;99 A4 > sec A

Some Sol!ed 50amples on complementary angles in trigonometry B14 5!aluate B cos "P 9 sin E" 9Solution Bcos "P 9sin E" 9> cos3 ;9 Q E" 4sin E" > sin E" 9sin E" 9> 1

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR4 Show that B 3 cos P9 94 3sin 9 94 O 3cos E; 94 sin "1 9 < sin "9 9> 9Solution B Consider


10/20

3 cos P9 94 3sin 9 94 O 3cos E; 94 sin "1 9 < sin "9 9> cos 3 ;9 9 9 94 sin 9 9 O cos3;9 9 "1 94 sin "1 9 < 0314 > sin 9 9 sin 9 9O sin "1 9 sin "1 9 < 0 1 #> 1 O 1 Q > 9

)roof of the Trigonometric identities

)roof. According to the )ythagorean theorem,

xOy> r. . . . . . . . . . . . . . . .314

Therefore, on di!iding $oth sides $y r,

x

2

r2 O

y2

r2 >1

That is, according to the definitions,

cosO sin > 1. . . . . . . . . . . . . .34

Apart from the order of the terms, this is the first trigonometric

&n di!iding line 34 $y cos, we ha!e

That is,

1 O tan> sec.

And if we di!ide $y sin, we ha!e
http://www.themathpage.com/atrig/unit-circle.htm#defshttp://www.themathpage.com/atrig/unit-circle.htm#defs


11/20

That is,

1 O cot

> cosec

..


12/20

&$'ecti!e B Dinding trigonometric ratios for "99,#E9and F99

Uaterial equiredB )astel sheet, geometry $o0

)rocedureB

A 1. Vraw an isosceles right triangle on a colorful paper and paste it onpastel sheet as shown

. Calculate the length of hypotenuse using )ythagoras theorem.

". 6CA > 6AC > #E9

#. @sing the triangle determine the !alue of Sin#E9, Cos#E9and Tan#E9

6. 1. Vraw an equilateral triangle on a colorful paper and paste it on pastelsheet as shown

A

CTIVIT


13/20

. Vraw )T perpendicular to W

". WT > T > cm

#.Dind )T using )ythagoras theorem

E. Calculate T)W

F. @sing )TW, determine the !alue for

SinF99, Cos F99 and Tan F99

Vetermine the Sin"99, Cos "99and Tan "99using the same triangle

&$ser!ationB

Sin#E9 >1L , Cos#E9 >1L and Tan#E9> 1

SinF99 >L", CosF99 >1 and TanF99> L"

Sin"99 >1 , Cos"99 >L"and Tan"99>1 L"

u$rics for ecording acti!ity work 31E marks4

)arameters Uarks allotted

6ring material foracti!ity

1

Takes interest in class 1


14/20

Is regular 1

%isten , o$ser!esattenti!ely

1

Takes care of property 1

G)erformance ofActi!ity

19

CompleteandCorrecttask

eeds helpto completetask

Independently works$utincompletetask

Tries tomakeeffort $utincompletetask

ust $eginthe task

19 < F E "

1. 2i!e some e0amples for o$'ects you find around you which are in the

shape of a right angled triangle.

. 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 aright angled triangle

!ecapitulat

on

%5Y5% 1

1. If triangle A6C is right angled at 6 and A6 > 1 cm , 6C > E cm ,

Assignmen


15/20

Dind 3I4 Sin A , Cot A. 3ii4 Cosec C, Cos C.

. If cot A>3

4 , find all other trigonometric ratios of the angle A.

". If E cos A Q 1 sin A > 9 ,find the !alue of

sinA+cosA

2COSAsinA

#. Simplify B 3 1 O tan 4 31 Q sin 4 31 O sin 4

E. 5!aluateB SinF99. Cos"99 O Sin "99. Cos F99

F. If A > F99 and 6 > "99!erify that Cos 3A64 > CosA Cos6 O SinA Sin6

P. )ro!e that B 3 1O Cot 43 1 Q Cos 431 O Cos 4 > 1

1 and Cos 3A Q64 >1 Dind A and 6.

;. 2i!en tanA>5

4 , 399Z A Z ;9 94, find the !alue of 1 sin A cos A.

19.If sec A > , find the !alue of cotA Q 1.

%5Y5%

1. If E cos A Q 1 sin A > 9 ,find the !alue ofsinA+cosA

2COSAsinA

. If tan A > 2 1 , show thattanA

1+ tan2

A >2

4

". If cot >3

4 , pro!e that cosec 2cot2

sec2

1>

73

#. If tan A > M, then pro!e that M sin A cos A > 1E

E. If cotA >#" c heck 1 Q tanA > cotA Q sinA


16/20

1 O tanA

F. An equilateral triangle is inscri$ed in a circle of radius F cm. Dind its

side.

P. Dind the !alue of Tan F99 geometrically.

1, Show that Cos O Sin >1 or 1

;. )ro!e that Tan 19Tan 9Tan "9[[. Tan 1

19 )ro!e that 3 Sin A O Sec A4O 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.

. If Cos Sin > 0 and Cos " Sin > y pro!e that

0 O yQ 0y > E

". If Sec O Tan > p, pro!e that Sin >p

2+1

p2

1

#.Compare the area of the right angled triangles A6C and V5D in which

A > "99 , 6 >;99, AC >#cm, V > F99 , 5 >;99 and

V5> #cm

E.If tan K O sin K > m and tan K sin K > n ,show that

3mQ n4> 1F mn

F. )ro!e that 3cosec K Q sinK4 3sec K Q cosK4 >1

tan+cot

P. )ro!e that

sin K 31O tanK4 O cos K31OcotK4> sec K OcosecK


17/20

;. )ro!e that

secK sec#K cosecK O cosec#K > cot#K tan#K

19. )ro!e that

SinF

A O CosF

A > 1 " Sin

A Cos

A

)&SSI6%5 5&S and their 5U5VIATI&S

1 7riting Sin ;9 instead of Sin 3;9 4 It has to $e emphasiHed that

for all complementary angles it has to $e written Sin 3;9 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.

50sin

cos > tan 5mphasiHe on writing ratio with correct angles and gi!e

sufficient practice.

# Students consider SinKO cosK > 1 and take its square root as Sin K O cos

K > \1 Instruct the students that square 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

emediation


18/20

questions Students should $e encouraged to use trigonometric results in

geometry, especially where the ratio of sides is gi!en.

To o$ser!e a relation $etween sum of squares of sine and cosine ratios of an

angle in a right angled triangle.

U5T8&VB 1.Consider three right angled triangles as shown

8ands on

acti!ity


19/20

. ecords the results in the ta$le $elow

ABC Sin C > Cos C > SinC OCosC>

]V5D Sin D > Cos D > SinD OCosD >

] )W Sin > Cos > Sin OCos>

&$ser!ationB

Sin O Cos > 1

Similarly we can show that

1 O tan > Sec . And 1 O Cot> Cosec

httpBwww.askmath.com

httpBwww.themathpage.com

www.pro'ectmaths

7e$

esources
http://www.ask-math.com/http://www.themathpage.com/http://www.projectmaths/http://www.ask-math.com/http://www.themathpage.com/http://www.projectmaths/


20/20

lesson plan trigo class x by manisha

Documents