If sin(A-B)=1/√10, cos(A+B) = 2/√29, find tan2A where A and B lie between 0 to pi by 4.

Solution:

We can proceed step by step by using trigonometric formulas to solve this problem.

By observing the given data we can use sin (A + B) and cos (A - B) formulae

Once we get the value for sin (A + B) and cos (A - B), we can easily find the tan 2A value

sin (A + B) = 1 - √ cos² (A+B)

sin (A + B) = √(1)² – (2/√29)² = √(1 - 4/29) = √25/29 = 5/√29 

sin (A - B) = 1/√10

cos (A - B) = 1 - √ sin²(A-B)

cos (A - B) = √(1)²  - 1/(√10)²  =√1  - 1/10 =  √9/10 = 3/√10

Using both the results we get the value for tan (A + B) and tan (A - B)

tan (A + B) = sin(A + B)/ cos (A + B)

tan (A + B) = 5/2

tan (A - B) = sin(A - B)/ cos (A - B)

tan (A - B) = 1/3

2A can also be written as A + B + A - B

Using above result to get the value for tan 2A  

tan2A = tan ((A + B) + (A - B))

= (5/2 + 1/3)/1 – (5/2)(1/3)

= 17

So, If sin(A-B)=1/√10, cos(A+B) = 2/√29,  tan2A is 17 where A and B lie between 0 to pi by 4

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If sin(A-B)=1/√10, cos(A+B) = 2/√29, find tan2A where A and B lie between 0 to pi by 4.

Summary:

If sin(A-B)=1/√10, cos(A+B) = 2/√29,  tan2A is 17 where A and B lie between 0 to pi by 4