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 1113583: Determine the constants a and b so that +%28-3%2B4%2Acos%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29+=+a%2Bb%2Asin%28x%29+ for all values of x.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%28-3%2B4%2Acos%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29+=+a%2Bb%2Asin%28x%29+

first simplify:

%28-3%2B4%2Acos%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29=+a%2Bb%2Asin%28x%29

%28-3%2B4%2A%281-sin%5E2%28x%29%29%29%2F%281-2%2Asin%28x%29%29=+a%2Bb%2Asin%28x%29

%28-3%2B4-4sin%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29=+a%2Bb%2Asin%28x%29

%281-4sin%5E2%28x%29%29%2F%281-2%2Asin%28x%29%29=+a%2Bb%2Asin%28x%29

%28%281-2sin%28x%29%29%281%2B2sin%28x%29%29%29%2F%281-2%2Asin%28x%29%29=+a%2Bb%2Asin%28x%29



2+sin%28x%29+%2B+1+=+a+%2B+b+sin%28x%29....solve for sin%28x%29

2sin%28x%29-+b+sin%28x%29+%2B+1+=+a+

%282-+b%29+sin%28x%29++=+a-1+

sin%28x%29=+%28a+-1%29%2F+%282+-+b%29

equal sin%28x%29 to 0

%28a+-1%29%2F+%282+-+b%29=0

if a-1=0 ->a=1
if 2-b=0-> b=2 (since denominator cannot be equal to zero)

so, your solutions are:
a+=+1
+b+=+2