document.write( "Question 869375: Find the minimum value of cosq + 2sinq, if q is an element of (-p/2, p/2)
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #524180 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "cosθ + 2sinθ\r\n" );
document.write( "\r\n" );
document.write( "We will strive to cause this to become a constant times\r\n" );
document.write( "the right side of the identity \r\n" );
document.write( "\r\n" );
document.write( "sin(α+β) = sinα·cosβ + cosα·sinβ\r\n" );
document.write( "\r\n" );
document.write( "Where β = θ  and \r\n" );
document.write( "\r\n" );
document.write( "sin(α+θ) = sinα·cosθ + cosα·sinθ\r\n" );
document.write( "\r\n" );
document.write( "We wish to find k and α so that \r\n" );
document.write( "\r\n" );
document.write( "cosθ + 2sinθ = k·sinα·cosθ + k·cosα·sinθ = k·sin(α+θ)\r\n" );
document.write( "\r\n" );
document.write( "So we must have 1 = k·sinα  and 2 = k·cosα\r\n" );
document.write( "\r\n" );
document.write( "or              sinα =   and cosα = \r\n" );
document.write( "\r\n" );
document.write( "Since  and \r\n" );
document.write( "\r\n" );
document.write( "we construct angle α in a right triangle by choosing α's\r\n" );
document.write( "opposite side to be 1 and its adjacent side to be 2,\r\n" );
document.write( "and its hypotenuse to be k.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "By the Pythagorean theorem, k² = 2²+1² = 4+1 = 5, so k = \r\n" );
document.write( "\r\n" );
document.write( "So\r\n" );
document.write( "\r\n" );
document.write( "cosθ + 2sinθ = k·sinα·cosθ + k·cosα·sinθ = k·sin(α+θ) \r\n" );
document.write( " \r\n" );
document.write( "becomes\r\n" );
document.write( "\r\n" );
document.write( "cosθ + 2sinθ = ·cosθ + ·cosα·sinθ = ·sin(α+θ) \r\n" );
document.write( "\r\n" );
document.write( "Since the amplitude of y = ·sin(α+θ) is , its\r\n" );
document.write( "\r\n" );
document.write( "minimum value is \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );