if tanx=1/5, tany=1/239 then the value of tan(4x-y)
Answers
Explanation:
Given :-
Tan x = 1/5
Tan y = 1/239
To find :-
Find the value of Tan (4x-y) ?
Solution :-
Given that
Tan x = 1/5 -------------(1)
Tan y = 1/239 ---------(2)
We know that
Tan 2A = 2 Tan A/(1-Tan² A)
Now,
Tan 2x = 2 Tan x /(1-Tan²x)
=> Tan 2x = 2(1/5)/[1-(1/5)²]
=> Tan 2x = (2/5)/[1-(1/25)]
=> Tan 2x = (2/5)/[(25-1)/25]
=> Tan 2x = (2/5)/(24/25)
=> Tan 2x = (2/5)×(25/24)
=> Tan 2x = (2×25)/(5×24)
=> Tan 2x = 50/120
=> Tan 2x = 5/12 --------(3)
Now,
We know that
Tan 4x = Tan 2(2x)
=> Tan 4x = 2 Tan 2x /(1-Tan²2x)
From (3)
=> Tan 4x = 2(5/12)/[1-(5/12)²]
=> Tan 4x = (10/12)/[1-(25/144)]
=> Tan 4x = (10/12)/[(144-25)/144]
=> Tan 4x = (10/12)/(119/144)
=> Tan 4x = (5/6)/(119/144)
=> Tan 4x = (5/6)×(144/119)
=> Tan 4x = (5×144)/(6×119)
=> Tan 4x = (5×24)/119
=> Tan 4x = 120/119 ---------(4)
Now
We know that
Tan (A-B) = (Tan A- Tan B)/(1+TanA Tan B)
Tan (4x-y) = (Tan 4x-Tan y)/(1+Tan 4xTan y)
On substituting these values in the above formula then
From (2) & (4)
=> Tan (4x-y)
=> [(120/119)-(1/239)]/[1+(120/119)(1/239)]
LCM of 119 and 239 = 28441
=> [{(120)(239)-(1)(119)}/28441]/[1+(120/28441)]
=>[(28680-119)/28441]/[(28441+120)/28441]
=> (28561/28441)/(28561/28441)
=> (28561/28441)×(28441/28561)
=> (28561×28441)/(28561×28441)
=> 1
Therefore, Tan(4x-y) = 1
Answer:-
The value of Tan(4x-y) for the given problem is 1
Used formulae:-
→ Tan 2A = 2 Tan A/(1-Tan² A)
→ Tan (A-B) = (Tan A- Tan B)/(1+TanA Tan B)
Explanation:
Given that
Tan x = 1/5
Tan y = 1/239
We know that
Tan 2A = 2 Tan A/(1-Tan² A)
Now,
Tan 2x = 2 Tan x /(1-Tan²x)
=> Tan 2x = = 2(1/5)/[1-(1/5)²]
=> Tan 2x = (2/5)/[1-(1/25)]
=> Tan 2x = (2/5)/[(25-1)/25] => Tan 2x = (2/5)/(24/25)
=> Tan 2x = (2/5)x(25/24)
=> Tan 2x = (2×25)/(5×24)
=> Tan 2x = 50/120
=> Tan 2x = 5/12 -(3)
Now,
We know that
Tan 4x = Tan 2(2x)
=> Tan 4x = 2 Tan 2x /(1-Tan²2x)
From (3)
=> Tan 4x = 2(5/12)/[1-(5/12)²]
=> Tan 4x = (10/12)/[1-(25/144)]
=
-(1)
-(2)
- U