Reciprocal relations between trigonometric ratios

                 Trigonometry - 6    

Hello friends ! trigonometric ratios aur trigonometric ratio se related question ke baare me jaanane ke baad aaj hum jaanege Relations between trigonometric ratios matlab Aaj hum trigonometric ratios ke beech kya sambandh hota hai ? Iske baare me hum details me jaanege.
Hum jaante hai ki, trigonometric ratios ke total 6 types hote hai. Jo iss prakar hai -:

1. sin θ
2. cos θ
3. tan θ
4. cosec θ
5. sec θ
6. cot θ      

Inn sabhi 6 trigonometric ratios ke beech me kya relations (sambandh) hai?
Jaise -:
• sin θ aur cos θ ke beech kya sambandh hai ?
• tan θ aur cot θ ke beech kya relation hai ?
• Issi prakar cosec θ aur sin θ, cos θ aur sec θ, cosec θ aur sin θ etc. ke beech kya relations hai ?
Aaj hum issi ke baare me jaanege.
So , bina time waste kiye aaj ka topic start karte hai-:

Relations between trigonometric ratios

Trigonometric ratios ke beech 3 important relations hai. Jo iss prakar hai -:

1. Reciprocal relation

2. Square relation

3. Relation among sinθ, cosθ, tanθ aur cotθ

1. Reciprocal relation-:

Iss relation ya sambandh ko hindi me व्युत्क्रम सम्बन्ध  ( vyutkram sambandh) ya उल्टा सम्बन्ध (ulta sambandh) kahate hai. Ab iss relations ko उल्टा सम्बन्ध (ulta sambandh) kyu kahte hai ? Ye mai aap niche bataunga.
Aayiye ab inn relations ko dekhte hai. Ye relations kuch iss prakaar se hai:

(i)     sin θ = 1 / cosec θ
(ii) cosec θ = 1 / sin θ

(iii)   cos θ = 1 / sec θ
(vi)   sec θ = 1 / cos θ

(v)  tan θ = 1 / cot θ

(vi)  cot θ = 1 /  tan θ
To ab aap inn relations ko dekh chuke . Ab mai aapko batata hu ki , iss relations ko उल्टा सम्बन्ध (ulta sambandh) kyu kahte hai ?

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Iska answer mai ek relation ko lekar batata hu.
sin θ = 1 / cosec θ
Upar diye relation me aap dekh sakte hai ki , sin θ aur 1/cosec θ diya hai .
Iska matlab ye huaa ki ,
Agar sin θ = 2/3 hai to cosec θ = 3/2 hoga. Ya agar cosec θ = 2/3 hai to sinθ = 3/2 hoga. Matlab ki , sin θ aur cosec θ dono ke beech उल्टा सम्बन्ध (ulta sambadh) hai.
Matlab , ye huaa ki -:
• sinθ ka ulata cosecθ
• cosecθ ka ulata sinθ
• cosθ ka ulata secθ
• secθ ka ulata cosθ
• tanθ ka ulata cotθ
• cotθ ka ulata tanθ

Isliye iss relation ko उल्टा सम्बन्ध (ulata sambandh) kahate hai. Ab uppar diye gaye relations se hume kuch aur relations milate hai. Jo iss prakar hai.
From (i) ,
(a)  sin θ × cosec θ = 1
Issi prakar (ii),(iii),(iv),(v) and (vi) ki help se -:
(b)  cos θ × sec θ = 1

(c)  tan θ × cot θ = 1
Ab mai aapko reciprocal relation  ko proof karke dikhata hu.
Suppose karte hai ki , ∆ABC ek Right angle triangle  hai , jisme /_CAB = θ   hai.

Figure-1

(i)     sin θ = 1 / cosec θ    and (ii) cosec θ = 1 / sin θ
Proof-:  Figure-1 ki help se , Hum jaante hai ki ,
sin θ = lamb/karn
          = BC/AC       .......(i)
cosec θ = Karn/lamb
              = AC/BC  ........(ii)
From (i) , sin θ = 1 / (AC/BC)
                            = 1/ cosec θ
⇒    sin θ = 1/ cosec θ
From (ii) , cosec θ = 1/ (BC/AC)
                                 = 1/sin θ
⇒    cosec θ = 1/sin θ



(iii)   cos θ = 1 / sec θ      and 
(vi)   sec θ = 1 / cos θ
Proof-: Figure-2 ki help se , Hum jaante hai ki ,
cos θ = Aadhar/karn
          = AB/AC       .......(i)
sec θ = Karn/Aadhar
              = AC/AB  ........(ii)
From (i) , cos θ = 1 / (AC/AB)
                            = 1/ sec θ
⇒    cos θ = 1/ sec θ
From (ii) , sec θ = 1/ (AB/AC)
                                 = 1/ cos θ
⇒    sec θ = 1/ cos θ

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(v)  tan θ = 1 / cot θ     and

(vi)  cot θ = 1 /  tan θ
Proof-: Fig.-2 ki help se , hum jaante hai ki -:
tan θ = lamb/Aadhar
          = BC/AB      .......(i)
cot θ = Aadhar/Lamb
          = AB / BC    ........(ii)
From (i) , tan θ = 1 / (AB/BC)
                            = 1/ cot θ
⇒    tan θ = 1/ cot θ
From (ii) , cot θ = 1/ (BC/AB)
                                 = 1/ tan θ
⇒    cot θ = 1/tan θ

How to learn easily reciprocal relations of trigonometric ratio ?

( Trigonometric ratio ke reciprocal relations ko aashani se kaise yaad kare ?)

Iss relations ko jaanane ke baad ab aapk mind me ye question aa raha hoga ki , inn relations ko yaad kaise rakhenge ? 

Aap isski bhi tension naa lo . Mai issk liye bhi aapko easily method batata hu . Jisase aap iss relations ko aashani se yaad rakh sakte hai.

Iss relation ko aashani se yaad rakhane ka method jaanane ke liye niche diye video ko dekhiye.

Watch this video-:

Trigonometry ke iss part me bas itana hi. Next post me mai aapko 2nd and 3rd relation ke baare me bataunga.


I hope ki , aapko aaj ka topic achha laga hoga. Agar aapko koi problem ho rahi ho to aap hume comment kark puch sakte hai.
Ye topic pasand aaya to please isse apne friends ke saath share kark humari help kare.
Thanks for reading !!!!

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