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}}}
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Question 1151636: Consider the following trigonometric equation:
A) It has the same solutions as the equation
B) the equation has no real solutions
C) It has the same solutions as the equation
D) It has the same solutions as the equation
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) (Show Source): You can put this solution on YOUR website!
Plug in A = sin(x) and B = cos(x)
The portion in red
is equivalent to 1 due to the pythagorean trig identity
---------------------------------------------------
Add C to both sides
We'll let
Plug in
Distribute
Combine like terms and do a bit of rearranging.
The RHS (right hand side) matches the original expression
So, is equivalent to 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
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.
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