document.write( "Question 1172407: Given constants A,w,a, where A and w are both positive. Show the function f(x)=Asin(wx+a) has period 2pi/w. \n" ); document.write( "

Algebra.Com's Answer #797423 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "Definition: A function f(x) is periodic if and only if f(x+T) = f(x) for all x in the domain. The T refers to the period.\r
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\n" ); document.write( "\n" ); document.write( "We claim the period is T = 2pi/w\r
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\n" ); document.write( "\n" ); document.write( "Let's compute f(x+T) which is the same as f(x+2pi/w)
\n" ); document.write( "Let's try to rewrite f(x+2pi/w) into f(x)\r
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\n" ); document.write( "\n" ); document.write( "f(x) = A*sin(wx+a)
\n" ); document.write( "f(x+2pi/w) = A*sin(w*(x+2pi/w)+a) ... replace every x with x+2pi/w
\n" ); document.write( "f(x+2pi/w) = A*sin(w*x+w*2pi/w+a) ... distribute
\n" ); document.write( "f(x+2pi/w) = A*sin(w*x+2pi+a)
\n" ); document.write( "f(x+2pi/w) = A*sin(w*x+a+2pi)
\n" ); document.write( "f(x+2pi/w) = A*sin((wx+a)+(2pi))
\n" ); document.write( "f(x+2pi/w) = A * [ sin(wx+a)*cos(2pi)+cos(wx+a)*sin(2pi) ] .... see note below
\n" ); document.write( "f(x+2pi/w) = A * [ sin(wx+a)*1+cos(wx+a)*0 ] .... use unit circle
\n" ); document.write( "f(x+2pi/w) = A*sin(wx+a)
\n" ); document.write( "f(x+2pi/w) = f(x)
\n" ); document.write( "f(x+T) = f(x)\r
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\n" ); document.write( "\n" ); document.write( "note: I used the identity sin(x+y) = sin(x)cos(y)+cos(x)sin(y)\r
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\n" ); document.write( "\n" ); document.write( "Since we've shown that f(x+2pi/w) = f(x) is true for all x in the domain, this verifies that f(x) is periodic with period T = 2pi/w
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