Rewriting sin &#x2061;<!-- ⁡ --> (

sweetymoeyz

sweetymoeyz

Answered question

2022-06-30

Rewriting sin ( a + b ) = c

Answer & Explanation

Dobermann82

Dobermann82

Beginner2022-07-01Added 15 answers

In general sin ( x ) = y is equivalent to
x = π n + ( 1 ) n arcsin ( y )  for some  n Z
So if x = π n + ( 1 ) n arcsin ( y )  for some  n Z then there exists an integer n such that
(1) a = π n + ( 1 ) n arcsin ( c ) b
This is equivalent to the WolframAlpha solution but is more compact.
Note that
sin ( π n + ( 1 ) n arcsin ( y ) ) = sin ( π n ) cos ( ( 1 ) n arcsin ( y ) ) + cos ( π n ) sin ( ( 1 ) n arcsin ( y ) ) = 0 + ( 1 ) n ( 1 ) n sin ( arcsin ( y ) ) = y

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