How does a sin ⁡ ( a − b ) become 2 a sin ⁡ a − b 2 cos ⁡ a − b 2 ?

Question

How does   a  sin  ⁡  (  a  −  b  ) become   2  a  sin  ⁡            a      −      b        2    cos  ⁡            a      −      b        2  ?

kliersel12g · Accepted Answer

Use the double angle formula for   sin, which states that   sin  ⁡  (  2  x  )  =  2  sin  ⁡  (  x  )  cos  ⁡  (  x  ) for all values of x. By setting   x  =  (  a  −  b  )      /    2, we see that  sin  ⁡  (  a  −  b  )  =  sin  ⁡      (    2    ×                  a        −        b            2        )    =  2  sin  ⁡      (                  a        −        b            2        )    cos  ⁡      (                  a        −        b            2        )      .If we multiply both sides of this equation by a, we obtain the desired result.

How does a sin(a−b) become 2a sin (a−b)/2 cos (a−b)/2?

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