Trigonometry - 7
Reciprocal relations ko batane ke baad aaj mai aapko square relation of trigonometric ratios ke baare me bataunga.
Ye relation bahut hi jyada useful aur important relation hai. Iss relation ko trigonometry me mostly use kiya jaata hai. Agar aap trigonometry ke question solve karte honge to iss relation ka use jarur kiye honge.
Reciprocal relations ko batane ke baad aaj mai aapko square relation of trigonometric ratios ke baare me bataunga.
Ye relation bahut hi jyada useful aur important relation hai. Iss relation ko trigonometry me mostly use kiya jaata hai. Agar aap trigonometry ke question solve karte honge to iss relation ka use jarur kiye honge.
Square relation ko hum ek dam basic se jaanege. Taki aapko iss topic se koi problem , pareshani ya doubt naa rahe . But phir bhi aapse request hai ki , aapko kahi par bhi problem ya doubt ho to jarur puchiye . Ab Aap kahenge kaise puchhu ?? To mai kahunga niche diye comment box , Facebook page massage , contact form ki help se.
To aayiye jyada time waste na karte huye aaj ka topic start karte hai -:
To aayiye jyada time waste na karte huye aaj ka topic start karte hai -:
Square Relation of trigonometric ratios
Iss relation ko hindi me varg sambandh ( वर्ग सम्बन्ध ) kahate hai. Ab mai aapko ye bata du ki ye relation hota kya hai ? Matalab kya hai ? iss relation ka.
Also read-:
• Quadratic polynomial ko binomial se fast and easily divide kaise karte hai ?
• Top-10 math tricks post November 2016
Also read-:
• Quadratic polynomial ko binomial se fast and easily divide kaise karte hai ?
• Top-10 math tricks post November 2016
What is a square relations of trigonometric ratios ??
Hum jaante hai ki trigonometric ratios 6 types ke hote hai. Unn sabhi ka square ( varg ) kariye. Ab square karne ke baad jo inn trigonometric ratios ke beech relations hoga ussi relations ko hum square relations kahate hai.
For example-: Maan lete hai ki , humare paas two trigonometric ratios sin θ aur cos θ hai. Ab inka square kariye-: sin²θ aur cos² .
Ab sin²θ aur cos²θ ke beech jo relation hoga ussi relation ko hum square relation kahate hai.
For example-: Maan lete hai ki , humare paas two trigonometric ratios sin θ aur cos θ hai. Ab inka square kariye-: sin²θ aur cos² .
Ab sin²θ aur cos²θ ke beech jo relation hoga ussi relation ko hum square relation kahate hai.
I hope ki , aapko samjh aaya hoga. Aayiye ab iss type ke sabhi relations ko dekhate hai.
Square relation ke 3 main relation iss prakar hai -:
(1) sin² θ + cos² θ = 1
(2) sec² θ - tan² θ = 1
(3) cosec² θ - cot² θ = 1
Ab inn 3 main relations ke sub-relations iss prakar hai -:
From (1) ,
(i) sin² θ = 1 - cos²θ
Or (ii) cos² θ = 1 - sin² θ
From (2),
(iii) sec² θ = 1 + tan² θ
Or (iv) tan² θ = sec² θ - 1
From (3),
(v) cosec² θ = 1 + cot² θ
Or (vi) cot² θ = cosec² θ - 1
Aayiye inn relation ko derived (nigaman ) karte hai -:
Suppose karte hai ki , Figure-1 me ∆ABC ek Right angle triangle hai. Jisme /_ABC = 90° hai.
According to pythagoras theorem -:
AC² = AB²+BC²
Or BC² + AB² = AC² ......(i)
Ab eq(i) ke dono side AC² se divide karne par-:
BC² + AB² = AC²
AC² AC² AC²
Or
(BC/AC)² + (AB/AC)² = 1 ......(ii)
Ab figure-1 ki help se -:
sin θ = BC/AC .....(*)
cos θ = AB/AC .....(**)
Ab (*) aur (**) ki value eq(ii) me rakhne par-:
(sin θ)² + (cos θ)² = 1
⇒ sin²θ + cos²θ = 1
⇒ sin²θ + cos²θ = 1
{(1) proof}
Also read-:
• square root in few seconds
Again eq(i) ke dono side BC² se divide karne par-:
BC² + AB² = AC²
BC² BC² BC²
Or
1 + (AB/BC)² = (AC/BC)² ......(iii)
Ab figure-1 ki help se -:
cot θ = AB/BC .....(*)
cosec θ = AC/BC .....(**)
Ab (*) aur (**) ki value eq(iii) me rakhne par-:
1 + (cot θ)² = (cosec θ)²
⇒ 1 + cot²θ = cosec²θ
⇒ cosec²θ - cot²θ = 1
{ (3) proof }
Again eq(i) ke dono side AB² se divide karne par-:
BC² + AB² = AC²
AB² AB² AB²
Or
(BC/AB)² + 1 = (AC/AB)² ......(iv)
Ab figure-1 ki help se -:
I hope ki , aapko aaj ka topic achhe se samjh aaya hoga. Aagar kahi par bhi aapko koi problem ya doubt ho to without waste your time abhi comment kark hum se puchiye. Hum puri koshish karenge ki aapki problem ya doubt dur ho jaaye.
Agar aapko ye topic pasand aaya aur aapko lagata hai ki , iss topic se kisi ki help ho sakti hai ya iss topic ki kisi aur ko bhi jarurat hai to please share kariye.
Thanks for reading !!!
Also read -:
• Fast multiply by 11 of any numbers in hindi
• Fast calculate square of numbers ending with 5
From (1) ,
(i) sin² θ = 1 - cos²θ
Or (ii) cos² θ = 1 - sin² θ
From (2),
(iii) sec² θ = 1 + tan² θ
Or (iv) tan² θ = sec² θ - 1
From (3),
(v) cosec² θ = 1 + cot² θ
Or (vi) cot² θ = cosec² θ - 1
Aayiye inn relation ko derived (nigaman ) karte hai -:
Suppose karte hai ki , Figure-1 me ∆ABC ek Right angle triangle hai. Jisme /_ABC = 90° hai.
 |
| Figure-1 |
According to pythagoras theorem -:
AC² = AB²+BC²
Or BC² + AB² = AC² ......(i)
Ab eq(i) ke dono side AC² se divide karne par-:
BC² + AB² = AC²
AC² AC² AC²
Or
(BC/AC)² + (AB/AC)² = 1 ......(ii)
Ab figure-1 ki help se -:
sin θ = BC/AC .....(*)
cos θ = AB/AC .....(**)
Ab (*) aur (**) ki value eq(ii) me rakhne par-:
(sin θ)² + (cos θ)² = 1
⇒ sin²θ + cos²θ = 1
⇒ sin²θ + cos²θ = 1
{(1) proof}
Also read-:
• square root in few seconds
Again eq(i) ke dono side BC² se divide karne par-:
BC² + AB² = AC²
BC² BC² BC²
Or
1 + (AB/BC)² = (AC/BC)² ......(iii)
Ab figure-1 ki help se -:
cot θ = AB/BC .....(*)
cosec θ = AC/BC .....(**)
Ab (*) aur (**) ki value eq(iii) me rakhne par-:
1 + (cot θ)² = (cosec θ)²
⇒ 1 + cot²θ = cosec²θ
⇒ cosec²θ - cot²θ = 1
{ (3) proof }
Again eq(i) ke dono side AB² se divide karne par-:
BC² + AB² = AC²
AB² AB² AB²
Or
(BC/AB)² + 1 = (AC/AB)² ......(iv)
Ab figure-1 ki help se -:
tan θ = BC/AB .....(*)
sec θ = AC/ AB .....(**)
Ab (*) aur (**) ki value eq(iv) me rakhne par-:
(tan θ)² + 1 = (sec θ)²
⇒ tan²θ + 1 = sec²θ
⇒ sec²θ - tan²θ = 1 { (2)proof }
sec θ = AC/ AB .....(**)
Ab (*) aur (**) ki value eq(iv) me rakhne par-:
(tan θ)² + 1 = (sec θ)²
⇒ tan²θ + 1 = sec²θ
⇒ sec²θ - tan²θ = 1 { (2)proof }
Agar aapko ye topic pasand aaya aur aapko lagata hai ki , iss topic se kisi ki help ho sakti hai ya iss topic ki kisi aur ko bhi jarurat hai to please share kariye.
Thanks for reading !!!
Also read -:
• Fast multiply by 11 of any numbers in hindi
• Fast calculate square of numbers ending with 5
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