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document.write( "Find the largest possible domain and largest possible range of the function
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document.write( "𝑔(𝑥) = 4 cos(3𝑥) − 3 sin(3𝑥).
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document.write( "Give your answers in set/interval notations.
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document.write( "The domain is, OBVIOUSLY, the set of all real numbers, since this function (this expression) \r\n" );
document.write( "is defined over all this set.\r\n" );
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document.write( "To find the range, let's make this identical transformation\r\n" );
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document.write( " 4*cos(3x) - 3*sin(3x) = 
. (1)\r\n" );
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document.write( "Next, notice that 
+ 
= 
= 
= 1.\r\n" );
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document.write( "THEREFORE, there is such angle 
that 
= 
, 
= 
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document.write( "This is simply the angle in QI, which satisfies this equation 
= 
, or = 
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document.write( "Then we can continue the equality (1) this way\r\n" );
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document.write( " 4*cos(3x) - 3*sin(3x) = = 
= \r\n" );
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document.write( " now apply the formula for sine of the sum of arguments \r\n" );
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document.write( " = 
.\r\n" );
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document.write( "Thus we presented the original expression as the sine function with amplitude 5 of argument 
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document.write( " 4*cos(3x) - 3*sin(3x) = .\r\n" );
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document.write( "It tells you that the range of 4*cos(3x) - 3*sin(3x) is the interval from -5 to 5, or, in the interval form, [-5,5].\r\n" );
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document.write( "ANSWER. The domain of the given function is the entire number line (-oo,oo).\r\n" );
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document.write( " The range of the given function is the interval [-5,5].\r\n" );
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document.write( "Solved.\r
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document.write( "This transformation and the logic, which I used, may seem as a focus - pocus.\r\n" );
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document.write( "But actually, it is a general transformation of the expression a*cos(x) - b*sin(x) with real coefficients \"a\" and \"b\"\r\n" );
document.write( "into single harmonic function\r\n" );
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document.write( " a*sin(x) - b*cos(x) = 
= \r\n" );
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document.write( " = 
= \r\n" );
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document.write( " = 
= 
.\r\n" );
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document.write( "where = 
.\r\n" );
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document.write( "It works always for any real coefficients \"a\" and \"b\" and transforms any linear combination a*cos(x) + b*sin(x) \r\n" );
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document.write( "into single harmonic function 
with the shift 
= and the amplitude 
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document.write( "It is very useful classic trigonometric transformation and the identity to know and to use in different \r\n" );
document.write( "trigonometric problems.\r\n" );
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document.write( "So, it makes sense to learn and to memorize it.\r\n" );
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