Question

(vii) cos 2e-cose = sino - sin20​

Answer 1

Answer:

Step-by-step explanation:

3.1 lexz voyksdu (Overview)

3.1.1 'kCn ‘trigonometry’ (f=kdks.kferh;) ;wukuh 'kCn ^fVªxksu*a (trigon) vkSj ^ehVªksu* (metron)

ls O;qRifÙk gqvk gS] ftldk vFkZ ,d f=kHkqt dh Hkqtkvksa dk ekiuk gSA ,d dks.k ,d fuf'pr js[kk

osQ lkis{k ifjHkze.k djus okyh fdlh js[kk osQ ?kw.kZu dh ek=kk gksrh gSA ;fn ;g ?kw.kZu nf{k.kkorZ

fn'kk esa gS rks dks.k ½.kkRed gksrk gS rFkk dks.k èkukRed gksrk gS] ;fn ?kw.kZu okekorZ fn'kk esa gksrk

gSA izk;%] ge dks.kksa dks ekius osQ fy,] nks izdkj dh i¼fr;k¡] vFkkZr~ (i) "kksf"Vd i¼fr

(sexagesinal system) vkSj (ii) o`Ùkh; i¼fr viukrs gSaA

"kkSf"Vd i¼fr esa] dks.k osQ ekiu dh bdkbZ va'k ;k fMxzh (Degree) gSA ;fn izkjafHkd Hkqtk

ls vafre Hkqtk rd dk ?kw.kZu ,d ifjHkze.k dk 1

360 ok¡ Hkkx gks] rks dks.k osQ eki dks 1° dgk tkrk

gSA bl i¼fr esa] oxhZdj.k fuEufyf[kr izdkj gSaµ

1° = 60′

1′ = 60″

ekiu dh o`Ùkh; i¼fr esa] ekiu dh bdkbZ jsfM;u (radian) gSA ,d jsfM;u og dks.k gS tks

fdlh o`Ùk dh f=kT;k osQ cjkcj yackbZ dk pki ml o`Ùk osQ osaQnz ij varfjr djrk gSA f=kT;k r okys

,d o`Ùk osQ pki PQ dh yackbZ s = rθ nh tkrh gS] tgk¡ θ jsfM;uksa esa ekik x;k og dks.k gS] tks

pki PQ o`Ùk osQ osaQnz ij varfjr djrk gSA

3.1.2 fMxzh vkSj jsfM;u esa lacaèk

fdlh o`Ùk dh ifjfèk dk mlosQ O;kl osQ lkFk lnSo ,d vpj vuqikr gksrk gSA ;g vpj vuqikr

π ls O;Dr dh tkus okyh ,d la[;k gS ftldk eku lHkh O;kogkfjd iz;kstu osQ fy, yxHkx 22

7

fy;k tkrk gSA fMxzh vkSj jsfM;u ekiksa osQ chp lacaèk fuEufyf[kr gSaµ

2 ledks.k = 180° = π jsfM;u

1 jsfM;u = 180°

π

= 57°16′ (yxHkx)

1° = 180

π jsfM;u = 0.01746 jsfM;u (yxHkx)

vè;k; 3

f=kdks.kferh; iQyu

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f=kdks.kferh; iQyu 35

3.1.3 f=kdks.kferh; iQyu

U;wu dks.kksa osQ fy,] f=kdks.kferh; vuqikr dks] fdlh ledks.k f=kHkqt dh Hkqtkvksa osQ vuqikrksa osQ

:i esa ifjHkkf"kr fd;k tkrk gSA jsfM;u eki esa O;Dr fdlh dks.k osQ fy,] f=kdks.kferh; vuqikr

dk foLrkj] f=kdks.kferh; iQyu dgykrk gSA f=kdks.kferh; iQyuksa osQ fofHkUu prqFkk±'kksa esa fpÉ

fuEufyf[kr rkfydk esa fn, gSaµ

I II III IV

sin x + + – –

cos x + – – +

tan x + – + –

cosec x + + – –

sec x + – – +

cot x + – + –

3.1.4 f=kdks.kferh; iQyuksa osQ izk¡r vkSj ifjlj

iQyu izkar ifjlj

sine R [–1, 1]

cosine R [–1, 1]

tan R – {(2n + 1)



2 : n ∈ Z} R

cot R – {nπ : n ∈ Z} R

sec R – {(2n + 1)



2 : n ∈ Z} R – (–1, 1)

cosec R – {nπ : n ∈ Z} R – (–1, 1)

3.1.5 ledks.k vFkkZr~ 90º ls NksVs ;k mlosQ cjkcj oqQN dks.kksa osQ sine, cosine vkSj

tangent

0° 15° 18° 30° 36° 45° 60° 90°

sine 0 6 2

4

− 5 1

4

− 1

2

10 2 5

4

− 1

2

3

2 1

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36 iz'u izn£'kdk

cosine 1 6 2

4

+ 10 2 5

4

+ 3

2

5 1

4

+ 1

2

1

2 0

tan 0 2 3 − 25 10 5

5

− 1

3 5 25 − 1 3

3.1.6 leoxhZ; ;k lacafèkr dks.k

dks.k 2

nπ ± θ leoxhZ ;k lacafèkr dks.k dgykrs gSa rFkk dks.k θ ± n × 360° lgkokluh

(coterminal) dks.k dgykrs gSaA O;kid leku;u osQ fy,] gesa fuEufyf[kr fu;e izkIr gSa%

( )

2

nπ ± θ osQ fy,] f=kdks.kferh; iQyu dk la[;kRed eku cjkcj gSµ

(a) mlh iQyu osQ eku osQ] ;fn n ,d le iw.kk±d gS rFkk bl eku dk fpÉ ml prqFkk±'k osQ

vuqlkj gksrk gS ftlesa og dks.k fLFkr gSA

(b) θ osQ laxr lgiQyu osQ eku osQ ;fn n ,d fo"ke iw.kk±d gS rFkk iQyu dk fpÉ ml prqFkk±'k

osQ vuqlkj gksrk gS] ftlesa og dks.k fLFkr gSA ;gk¡ sine vkSj cosine, tan vkSj cot rFkk sec

vkSj cosec ,d nwljs osQ lgiQyu gSaA