Graphing Sine and Cosine
Key Questions
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The domain of a function
f(x) are all the values ofx for whichf(x) is valid.The range of a function
f(x) are all the values whichf(x) can take on.sin(x) is defined for all real values ofx , so it's domain is all real numbers.However, the value of
sin(x) , its range , is restricted to the closed interval [-1, +1]. (Based on definition ofsin(x) .)Alan P. · 3 · Feb 27 2015
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The maximum value of the function
cos(x) is1 .This result can be easily obtained using differential calculus.
First, recall that for a function
f(x) to have a local maximum at a pointx_0 of it's domain it is necessary (but not sufficient) thatf^prime(x_0)=0 . Additionally, iff^((2)) (x_0)<0 (the second derivative of f at the pointx_0 is negative) we have a local maximum.For the function
cos(x) :d/dx cos(x)=-sin(x) d^2/dx^2 cos(x)=-cos(x) The function
-sin(x) has roots at points of the formx=n pi , wheren is an integer (positive or negative).The function
-cos(x) is negative for points of the formx= (2n+1) pi (odd multiples ofpi ) and positive for points of the form2n pi (even multiples ofpi ).Therefore, the function
cos(x) has all it's maximums at the points of the formx=(2n+1)pi , where it takes the value1 .
Vinicius M. G. Silveira
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4
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Dec 10 2014
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Since the function passes from the origin, in fact
sin0=0 ,
the y-intercept is0 .graph{sinx [-10, 10, -5, 5]}
Massimiliano
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2
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Mar 2 2015