SOLUTION: Determine the constants a and b so that {{{ (-3+4 cos^(2) x)/ (1-2 sin x) = a+b sin x }}} for all values of x.

Algebra ->  Trigonometry-basics -> SOLUTION: Determine the constants a and b so that {{{ (-3+4 cos^(2) x)/ (1-2 sin x) = a+b sin x }}} for all values of x.      Log On


   

Question 1192560: Determine the constants a and b so that +%28-3%2B4+cos%5E%282%29+x%29%2F+%281-2+sin+x%29+=+a%2Bb+sin+x+ for all values of x.
Answer by ikleyn(52798) About Me  (Show Source):
You can put this solution on YOUR website!
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Determine the constants a and b so that +%28-3%2B4+cos%5E%282%29+x%29%2F+%281-2+sin+x%29+=+a%2Bb+sin+x+ for all values of x.
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Transform the numerator step by step

    -3+%2B+4%2Acos%5E2%28x%29 = -3+%2B+4%2A%281-sin%5E2%28x%29%29 = -3+%2B+4+-+4%2Asin%5E2%28x%29 = 1+-+4%2Asin%5E2%28x%29 = %281-2sin%28x%29%29%2A%281%2B2sin%28x%29%29.


So, your fraction now is

    %28-3%2B4+cos%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29 = %28%281-2sin%28x%29%29%2A%281%2B2sin%28x%29%29%29%2F%281-2%2Asin%28x%29%29.


Cancel the factors ((1-2*sin(x)) in both the numerator and denominator.  You will get then

    %28-3%2B4+cos%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29 = %28%281-2sin%28x%29%29%2A%281%2B2sin%28x%29%29%29%2F%281-2%2Asin%28x%29%29 = 1 + 2*sin(x).


It is just the form which you need.  So,  a= 1,  b= 2.


ANSWER.  a= 1,  b= 2.

Solved.