SOLUTION: Consider the following trigonometric equation: {{{2sin(x)cos(x)-3sin(x)-3cos(x)+3=0}}} A) It has the same solutions as the equation {{{0=(sin(x)+cos(x))^2-3(sin(x)+cos(x))+2}}}

Algebra ->  Trigonometry-basics -> SOLUTION: Consider the following trigonometric equation: {{{2sin(x)cos(x)-3sin(x)-3cos(x)+3=0}}} A) It has the same solutions as the equation {{{0=(sin(x)+cos(x))^2-3(sin(x)+cos(x))+2}}}       Log On


   

Question 1151636: Consider the following trigonometric equation:
2sin%28x%29cos%28x%29-3sin%28x%29-3cos%28x%29%2B3=0
A) It has the same solutions as the equation 0=%28sin%28x%29%2Bcos%28x%29%29%5E2-3%28sin%28x%29%2Bcos%28x%29%29%2B2
B) the equation has no real solutions
C) It has the same solutions as the equation 0=%28sqrt%282%29sin%28x%2Bpi%2F4%29-2%29%28sqrt%282%29sin%28x%2Bpi%2F4%29-1%29
D) It has the same solutions as the equation 0=%28sqrt%282%29sin%28x%2Bpi%2F4%29%29%5E2-3%28sqrt%282%29sin%28x%2Bpi%2F4%29%29%2B2
E) none of these
Note: can you please show me how to find out whether the equation has a real solution or not.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

%28A%2BB%29%5E2+=+A%5E2+%2B+2%2AA%2AB+%2B+B%5E2

Plug in A = sin(x) and B = cos(x)



The portion in red

%28sin%28x%29%2Bcos%28x%29%29%5E2+=+highlight%281%29+%2B+2%2Asin%28x%29%2Acos%28x%29 is equivalent to 1 due to the pythagorean trig identity

%28sin%28x%29%2Bcos%28x%29%29%5E2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29

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%28sin%28x%29%2Bcos%28x%29%29%5E2+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29

%28sin%28x%29%2Bcos%28x%29%29%5E2+%2B+C+=+1+%2B+2%2Asin%28x%29%2Acos%28x%29+%2B+C Add C to both sides

We'll let C+=+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2

Plug in C+=+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2

Distribute

Combine like terms and do a bit of rearranging.

The RHS (right hand side) matches the original expression
So,%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2 is equivalent to 2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3 for all real numbers x.

Therefore, the final answer is choice A

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Through graphing technology, we can use programs like GeoGebra to visually verify we have the proper answer.

Let
f%28x%29+=+2%2Asin%28x%29%2Acos%28x%29-3sin%28x%29+-+3cos%28x%29%2B3
g%28x%29+=+%28sin%28x%29%2Bcos%28x%29%29%5E2+-+3%28cos%28x%29%2Bsin%28x%29%29%2B2
The graph below shows a red and blue striped curve. This is really a solid red curve underneath with a blue dashed curve directly on top of it.
f(x) = red solid curve, g(x) = blue dashed curve
The table of values show that f(x) = g(x) for any real x. While this isn't an exhaustive proof by any means, it's a quick way to verify the answer.

I recommend you play around with GeoGebra (or any equivalent graphing technology) to turn off/on the second graph to see how it is directly over top the first graph.