Solve Trigonometric Ratios: 2ab/(b^2 - a^2)

AI Thread Summary
The discussion focuses on finding the expression for tan(x+y) given the equations sin x + sin y = a and cos x + cos y = b. The initial attempts involve manipulating trigonometric identities but lead to confusion. Key formulas for sine and cosine sums are suggested to simplify the expressions. Ultimately, the poster successfully solves the problem and expresses tan(x+y) in terms of a and b as 2ab/(b^2 - a^2). The thread concludes with gratitude for the assistance received.
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Homework Statement


Given:
sin x + sin y= a
cos x + cos y=b
find tan(x+y)
Prove it to be equal to"
\frac{2ab}{b^{2}-a^{2}}


The Attempt at a Solution



I get stuck after this:
2ab= 2 sin (x+y)+sin 2x+sin 2y
b^{2}-a^{2}= cos 2x + cos 2y + 2 cos (x+y)

Division of the two never results in tan (x+y).
Plase help me express tan (x+y) in terms of a,b!
 
Last edited:
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Alright, I checked these two corrections and they look correct.
2ab= 2 sin (x+y)+sin 2x+sin 2y
b^{2}-a^{2}= cos 2x + cos 2y + 2 cos (x+y)
For the part,
sin(2x)+sin(2y)
you can use the sum and difference formulas for sines and cosines you will get
(3) 2 sin \left( \frac{2x+2y}{2} \right) cos \left( \frac{2x+2y}{2} \right) = 2sin(x+y)cos(x-y)

For the part, cos(2x)+cos(2y)
(4) 2 cos \left( \frac{2x+2y}{2} \right) cos \left( \frac{2x-2y}{2} \right)=2cos(x+y)cos(x-y)

Edited.
 
Last edited:
konthelion said:
Since, you know that
(1) sin x + sin y= a
(2) cos x + cos y=b

Then by the sum and difference formulas, you will get
(3) 2 sin \left( \frac{x+y}{2} \right) cos \left( \frac{x+y}{2} \right) = a
(4) 2 cos \left( \frac{x+y}{2} \right) cos \left( \frac{x-y}{2} \right) = b

No, wait scratch this. This won't help.

Thanks a lot friend. I hav solved it now. Thanks once again.
 
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