document.write( "Question 1043975: Express −5.9sin(x)+6.6cos(x) in the form ksin(x+ϕ)\r
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\n" ); document.write( "\n" ); document.write( "Note: ϕ should be given in radians in the interval −π<ϕ<π \n" ); document.write( "
Algebra.Com's Answer #659325 by Edwin McCravy(20077)\"\" \"About 
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document.write( "(1)     -5.9sin(x)+6.6cos(x) in the form ksin(x+ϕ)\r\n" );
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document.write( "That makes us think of the right side of the identity:\r\n" );
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document.write( "cos(x+ϕ) = cos(x)cos(ϕ) - sin(x)sin(ϕ)\r\n" );
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document.write( "multiplied by some constant k:\r\n" );
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document.write( "(2)     k*cos(x+ϕ) = k[cos(x)cos(ϕ) - sin(x)sin(ϕ)]\r\n" );
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document.write( "Let's get \r\n" );
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document.write( "(1)     -5.9sin(x)+6.6cos(x)\r\n" );
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document.write( "in the form of the right side of (2)\r\n" );
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document.write( "        k[cos(x)cos(ϕ) - sin(x)sin(ϕ)]  \r\n" );
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document.write( "(1)     -5.9sin(x)+6.6cos(x) =\r\n" );
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document.write( "        cos(x)(6.6) - sin(x)(5.9) \r\n" );
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document.write( "Let A be some positive non-zero constant. Then by multiplying\r\n" );
document.write( "then dividing by A\r\n" );
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document.write( "       \"A%28cos%28x%29%286.6%2FA%29%5E%22%22+-+sin%28x%29%285.9%2FA%29%29\" \r\n" );
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document.write( "We find A and ϕ such that\r\n" );
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document.write( "\"6.6%2FA=cos%28phi%29\" and \"5.9%2FA=sin%28phi%29\"\r\n" );
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document.write( "We see that ϕ must be in QIV because that's where the\r\n" );
document.write( "cosine is positive and the sine is positive.  We draw\r\n" );
document.write( "an angle in QI to represent ϕ.  \r\n" );
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document.write( "We can find A by the Pythagorean theorem:\r\n" );
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document.write( "\"A%5E2=%286.6%29%5E2%2B%28-5.9%29%5E2%29\"\r\n" );
document.write( "\"A%5E2=43.56%2B34.81\"\r\n" );
document.write( "\"A%5E2=78.37\"\r\n" );
document.write( "\"A=8.852683209\"\r\n" );
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document.write( "We can find ϕ from \r\n" );
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document.write( "\"tan%28phi%29=5.9%2F6.6\"\r\n" );
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document.write( "using the inverse tangent feature on a calculator\r\n" );
document.write( "in radian mode:\r\n" );
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document.write( "ϕ = 0.7294565922\r\n" );
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document.write( "So \r\n" );
document.write( "       \"A%28cos%28x%29%286.6%2FA%29%5E%22%22+-+sin%28x%29%285.9%2FA%29%29\" \r\n" );
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document.write( "now becomes\r\n" );
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document.write( "       \"A%28cos%28x%29cos%28phi%29+-+sin%28x%29sin%28phi%29%29\"\r\n" );
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document.write( "which is equivalent to\r\n" );
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document.write( "      \"A%2Acos%28x%2Bphi%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "      \"8.852683209%2Acos%28x%2B0.7294565922%29\"\r\n" );
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document.write( "Edwin
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