How would you solve the following equation? 1 + 2 sin &#x2061;<!-- ⁡ --> ( x ) =

Osmarq5ltp

Osmarq5ltp

Answered question

2022-05-13

How would you solve the following equation?
1 + 2 sin ( x ) = 2 sin ( x 2 ) + 2 cos ( x 2 ) for  2 π x 2 π .

Answer & Explanation

Corinne Choi

Corinne Choi

Beginner2022-05-14Added 15 answers

Solution
If we expand the brackets above we get:
sin 2 ( x ) + sin ( x ) + 1 4 = cos 2 ( x 2 ) + 2 cos ( x 2 ) sin ( x 2 ) + sin 2 ( x 2 )
sin 2 ( x ) + sin ( x ) + 1 4 = 1 + sin ( x )
sin 2 ( x ) = 3 4
This is the same as what you had but it is a coincidence. Your working is still incorrect (or at least very very brief and skipping some important working out).
sin ( x ) = ± 3 2
x = π 3 , 2 π 3 , 4 π 3 , 5 π 3  for  0 x 2 π
We need to substitute these back in and check each answer as taking the square root could introduce extraneous answers of which we find one. Hence we get:
x = π 3 , 2 π 3 , 5 π 3  for  0 x 2 π
Obviously there exists other solutions which are these plus integer multiple of 2 π

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