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Eliminate theta if tan(theta-alpha)=a and tan(theta+alpha)=b

Eliminate theta if tan(theta-alpha)=a and tan(theta+alpha)=b

Grade:11

2 Answers

Arun
25750 Points
5 years ago
We know that if -- 

Tan x = A 

=> x = Tan⁻¹ A 

Hence the given eqns can be re-written as -- 

.... θ - α = Tan⁻¹ a .................. (1) and also -- 
.... .θ + α = Tan⁻¹ b .................. (2) 
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-- 2 α = Tan⁻¹ a - Tan⁻¹ b ...................... On subtraction 

=> Tan⁻¹ a - Tan⁻¹ b = -- 2 α
tejas rajabhoj
15 Points
3 years ago
eliminate theta if tan( theta -alpha) =a and tan (theta alpha)=b​
Tan(θ - α)  = a
Tan(θ + α) = b
Using Tan(x + y)  = (Tanx + Tany)/(1 - TanxTany)
          Tan(x - y)  = (Tanx - Tany)/(1 + TanxTany)
Tan(θ - α)  =   (Tanθ - Tanα)/(1 + TanθTanα)  = a
=> Tanθ - Tanα = a  + aTanθTanα
=> Tanθ(1 + aTanα) = a + Tanα
=> Tanθ =  (a + Tanα)/(1 + aTanα)
Tan(θ + α) =  (Tanθ + Tanα)/(1 - TanθTanα) = b
=> Tanθ + Tanα = b  - bTanθTanα
=> Tanθ(1 + b Tanα) = b - Tanα
=> Tanθ =  (b - Tanα)/(1 + bTanα)
Equating Tanθ
(a + Tanα)/(1 + aTanα)  =  (b - Tanα)/(1 + bTanα)
=>  (a + Tanα)(1 + bTanα) =  (1 + aTanα) (b - Tanα)
=> a + Tanα + abTanα + bTan²α  =  b - Tanα  + abTanα  - aTan²α
=> 2Tanα  + (a + b)Tan²α =  (b + a)
=> 2 Tanα = (b+a)( 1  - Tan²α)
=> b + a =  2 Tanα/ ( 1  - Tan²α)
=> b + a = (Tan α + Tanα)/(1  - TanαTanα)
=> b + a = Tan(α + α)
=> b + a = Tan(2α)
 
 

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