Graphing Sine and Cosine
Key Questions
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The domain of a function
#f(x)# are all the values of#x# for which#f(x)# is valid.The range of a function
#f(x)# are all the values which#f(x)# can take on.#sin(x)# is defined for all real values of#x# , 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 of#sin(x)# .)Alan P. · 3 · Feb 27 2015
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The maximum value of the function
#cos(x)# is#1# .This result can be easily obtained using differential calculus.
First, recall that for a function
#f(x)# to have a local maximum at a point#x_0# of it's domain it is necessary (but not sufficient) that#f^prime(x_0)=0# . Additionally, if#f^((2)) (x_0)<0# (the second derivative of f at the point#x_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 form#x=n pi# , where#n# is an integer (positive or negative).The function
#-cos(x)# is negative for points of the form#x= (2n+1) pi# (odd multiples of#pi# ) and positive for points of the form#2n pi# (even multiples of#pi# ).Therefore, the function
#cos(x)# has all it's maximums at the points of the form#x=(2n+1)pi# , where it takes the value#1# .
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 is#0# .graph{sinx [-10, 10, -5, 5]}
Massimiliano
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2
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Mar 2 2015