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rjawbreakerca 2022-07-07 Answered
General solution to sin α + sin β and cos α + cos β?
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Answers (1)

Jamiya Costa
Answered 2022-07-08 Author has 18 answers
Using the sum/difference identities of the sine function, we find that
sin ( α + β ) = sin ( α ) cos ( β ) + sin ( β ) cos ( α )
sin ( α β ) = sin ( α ) cos ( β ) sin ( β ) cos ( α )
Adding them together, we find that
sin ( α + β ) + sin ( α β ) = 2 sin ( α ) cos ( β )
Now suppose that we want to find sin ( x ) + sin ( y ) where x<y. Then we can rewrite x and y as α ± β where α is the average of x and y and where β is the difference between y and the average. From there, we can plug it into the identity above. A similar computation can be arrived at using the sum/difference formulas for cosine.
α = x + y 2 , β = y x + y 2
x = α β , y = α + β
sin ( x ) + sin ( y ) = sin ( α β ) + sin ( α + β ) = 2 sin ( α ) cos ( β )

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