Question

Part 1: Using complete sentences, compare the key features and graphs of sine and cosine. What are their similarities and differences?

Part 2: Using these similarities and differences, how would you transform f(x) = 2 sin(2x - π) + 3 into a cosine function in the form f(x) = a cos(bx - c) + d?

Answer 1

@TheSmartOne

Answer 2

similarities?
their range is the same
their period is the same
they both repeat forever

Answer 3

differences?
the sine graph looks like a shifted version of the cosine graph

Answer 4

im not sure what your instructor is looking for EXACTLY. Im sure you would have learned what words to use in class

Answer 5

as for part two i'll get working on some graphs to show you

Answer 6

but first i'll tell you this
Acos(Bx-C)+D
A will change the amplitude
B will change the length of the period
C will change where the period starts
D will change the position of the cosine on the y axis

Answer 7

since cosine is just a shifted version of sine we want A, B, and D to be exactly the same

Answer 8

so lets start with 2cos(2x)+3

Answer 9

the two graphs look almost right

Answer 10

to shift the (red) cosine graph to look like the (blue) sine graph we have to change the C

Answer 11

since i know the difference between sine and cosine will always be some fraction of pi or multiple of pi i can make an educated guess

Answer 12

the distance LOOKS like 1 unit across
so i can guess that the REAL distance is pi/4 since that equals around 1

Answer 13

when shifting these kinds of graphs
DISTANCE SHIFTED = -C/2
in our case distance shifted = pi/4
pi/4 = -C/2
pi/2 = -C
-pi/2 = C

Answer 14

so C equals -pi/2
plug that back into our equation we get
2cos(2x-(-pi/2))+3
which equals
2cos(2x+pi/2)+3

Answer 15

since your question asks for there to be a negative sign there we can just do a little trick
\[Acos(Bx \color{red}-C)+D\]

Answer 16

adding or subtracting 2pi to C does nothing to the graph
so if i just subtract 2pi
\[\frac{ \pi }{ 2 }-2\pi=-\frac{ 3\pi }{ 2 }\]
our new C is -3pi/2

Answer 17

FINAL ANSWER TO PART 2
\[2\cos(2x-\frac{ 3\pi }{ 2 })+3\]