The graph of y = sin(x) is the same as the graph of y = cos(x) shifted to the ri

Question

The graph of y = sin(x) is the same as the graph of y = cos(x) shifted to the right /2 units, so the sine curve y = sin(x) is also at the same time a cosine curve: y = cos(x-). In fact, any sine curve is also a cosine curve with a different horizontal shift, and any cosine curve is also a sine curve. Sine and cosine curves are collectively referred to as sinusoidal 3x 2 For the curve whose qraph is shown, find all possible ways of expressing it as a sine curve y = a sin(x-b) or as a cosine curve y = a cos(x-b). (Let n be any integer.) y = 2 sin(x- y--2 sink- y 2 cos(x- y = -2 cos(x- Explain why you think you have found all possible choices for a and b in each case.

Explanation / Answer

amplitude of the graph is 2 units

so |a| =2

for sine graph :

when a>0

b=2n + , where n is an integer

when a<0

b=2n

y=2sin(x-(2n+))

y=-2sin(x- 2n)

for cosine graph :

when a>0

b=2n +(3/2) , where n is an integer

when a<0

b=2n+(/2)

y=2cos(x-(2n+(3/2)))

y=-2cos(x- (2n+(/2)))

all possible choices are found because the period of given graph is 2.

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