What is another form of cos(30◦-A)+ sin(60◦+A) ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that `sin alpha = cos(90^o - alpha)` , hence `sin (60^o + A) = cos(90^o - 60^o - A)`

`sin (60^o + A) = cos(30^o - A)`

Hence, substituting `cos(30^o - A)`  for `sin (60^o + A)`  in `cos(30^o - A) + sin (60^o + A)`  yields:

`cos(30^o - A) + sin (60^o + A) = cos(30^o - A) + cos(30^o - A)`

`cos(30^o - A) + sin (60^o + A) = 2cos(30^o - A)`

You need to expand `cos(30^o - A)`  such that:

`cos(30^o - A) = cos30^o*cos A + sin 30^o*sin A`

Substituting  `1/2`  for sin `30^o`  and `sqrt3/2`  for cos `30^o`  yields:

`cos(30^o - A) = sqrt3*cos A/2 + sin A/2`

Hence, multiplying by 2 both sides yields:

`2cos(30^o - A) = sqrt3*cos A + sin A`

Hence `cos(30^o - A) + sin (60^o + A) = sqrt3*cos A + sin A` .

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