How many solutions exist for the equation 2 sin &#x2061;<!-- ⁡ --> ( x ) +

Oakey1w

Oakey1w

Answered question

2022-06-13

How many solutions exist for the equation 2 sin ( x ) + cos ( x ) = 3 in [ 0 , 2 π ]?

Answer & Explanation

Braedon Rivas

Braedon Rivas

Beginner2022-06-14Added 24 answers

Hint: A sin ( x ) + B cos ( x ) = c sin ( x + x 0 )
In order to derive the new coefficients:
c sin ( x + x 0 ) = c sin ( x ) cos ( x 0 ) + c sin ( x 0 ) cos ( x ) = A sin ( x ) + B cos ( x )
Compare coefficients: A = c cos ( x 0 ) and A = c cos ( x 0 ). Divide the equations. B / A = tan ( x 0 ) and square and add both equations A 2 + B 2 = c 2 .
Hence, c = A 2 + B 2 = 5 and x 0 = arctan ( B / A ) = arctan ( 1 / 2 )

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