Q)tan(A + B)=1 and tan(A-B)=1/7 Find tanA and tan B?

Asked by yask12 | 1st Aug, 2010, 04:04: PM

Expert Answer:

Dear Student,
 
Here,  
 
tan(A+B) = 1

which means

tan A + tan B
______________  = 1
1 - tan A tan B

tan A + tan B = 1-tan A tan B
tan A + tan B - 1+ tan A tan B = 0


Also, tan(A - B) = 1/7
 
So,

tan A - tan B
_____________  = 1/7
1+tan A tan B

7tan A - 7tan B = 1 + tan A tan B
 
7tan A - 7tan B - 1 - tan A tan B = 0


tan A + tan B-1+tan A tan B=0

7tan A-7tan B-1-tan A tan B=0

Let
tan A = x and tan B = y

x + y - 1 + xy = 0
x + y + xy = 1  ...........(a)

7x - 7y - 1 - xy = 0
7x - 7y - xy = 1   ..........(b)


From equation (a)
 
x + y + xy = 1
x + xy = 1 - y
x(1 + y) = 1 - y
x = (1-y)/(1+y)

Replacing this value of x in equation (b), we get


7(1 - y)/(1 + y) - 7y - y(1 - y)/(1 + y) = 1

[(7-7y)/1+y] -7y - (y - y2)/(1+y) = 1

(7 - 7y)/(1 + y) - (y - y2)/(1+y)=1 + 7y

7 - 7y - y+y2
_____________  =1 + 7y
   1 + y

7 - 8y + y2 = (1 + 7y)(1 + y)
 
7 - 8y + y2 = 1 + 8y + 7y2
 
6y2 + 16y - 6 = 0
 
(6y - 2)(y + 3)=0

y = 1/3 or y = -3(rejected)

So, when y = 1/3
x = (1 - y)/(1+y)
x = (1 - 1/3)/(1+1/3)
x = 1/2

Hence,   tan A=1/2 and tan B=1/3
 
 
Regards,
 
Team Topper Learning

Answered by  | 1st Aug, 2010, 06:43: PM

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