Question 351524
Look at {{{A*sin(x+y)}}}
{{{A*sin(x+y)=A*cos(y)sin(x)+A*cos(x)sin(y)
}}}
{{{A*sin(4t+b)=A*cos(b)sin(4t)+A*cos(4t)sin(b)
}}}
Solve for {{{A}}} and {{{b}}}.
Comparing terms,
{{{A*cos(b)=5}}}
{{{A*sin(b)=2}}}
{{{tan(b)=(2/5)}}}
{{{b=atan(2/5)}}}
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If {{{tan(b)=2/5}}}, then {{{cos(b)=5/sqrt(2^2+5^2)=5/sqrt(29)}}}
{{{A*cos(atan(2/5))=5}}}
{{{A=5/cos(atan(2/5))=5/(5/sqrt(29))=sqrt(29)}}}
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{{{2 cos4t+ 5 sin4t=sqrt(29)*sin(4t+atan(2/5))}}}
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Amplitude : {{{A=sqrt(29)}}}
Period: {{{P=2*pi/(4)=pi/2}}}