SOLUTION: Hi can you please help me with these sinusoidal functions questions? Please include all answers in degree mode and not in radian. Thank you! 1. State the transformations that ha

Question 1202255: Hi can you please help me with these sinusoidal functions questions? Please include all answers in degree mode and not in radian. Thank you!
1. State the transformations that have been applied to y = cos x to obtain the function y = 1.78cos(x - 120°)+ 10
2.
3.
4. Determine the equation of a sine function that would have a range of {y∈R | -4 ≤ y ≤ 1} and a period of 45°. Determine the cosine function that results in the same graph.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52873) (Show Source): You can put this solution on YOUR website!
.

Of four tasks in this post, I will solve only ONE, number 1.

The transformations are, in this order - shift the plot right 120 units (the units are degrees, in this case); - stretch the plot vertically with the coefficient 1.78. - shift the plot 10 units up.

Solved.

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Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

I'll focus on problem 4 to get you started.

The template for the sine function is
y = A*sin(B(x-C))+D

|A| = amplitude
B = used to find the period
C = phase shift
D = midline

The next four sections will go over how to find the values for A,B,C,D

-----------------------------------------------

Determine the value of A.

Range is {y∈R | -4 ≤ y ≤ 1}
y = -4 = min
y = 1 = max
The vertical distance between these y values is 5 units.
Count the spaces, or subtract and use absolute value, to determine this distance. A vertical number line might help visualize things.
Half of this vertical distance is 5/2 = 2.5
We either can have A = 2.5 or A = -2.5
I'll go with A = 2.5

-----------------------------------------------

Determine the value of B.

We want a period of 45 degrees
T = period = 45
B = 360/T
B = 360/45
B = 8

-----------------------------------------------

Determine the value of C.

The phase shift can be whatever we want. Let's go for C = 0 as it's easiest.
There's not much else to this section.

-----------------------------------------------

Determine the value of D.

Range is {y∈R | -4 ≤ y ≤ 1}
y = -4 = min
y = 1 = max
Apply the midpoint formula to get (-4+1)/2 = -3/2 = -1.5
This is the value of D

-----------------------------------------------

Summary
A = 2.5
B = 8
C = 0
D = -1.5

We therefore go from
y = A*sin(B(x-C))+D
to
y = 2.5*sin(8(x-0))+(-1.5)
and it fully simplifies to
y = 2.5*sin(8x)-1.5
That is one possible sine equation that has a period of 45 degrees and has a range of -4 ≤ y ≤ 1 where y is a real number
I recommend using Desmos to graph out the function to verify the answer

Graph
https://www.desmos.com/calculator/9qhl97rerg

GeoGebra is another useful graphing app.

The name "cosine" heavily implies that we have a sinusoidal curve.
"sine" is buried in the name "cosine".
Cosine is a phase-shifted version of sine.
For more information, search out "trig co-functions".

To go from sine to cosine, we involve a phase shift of 90 degrees
This is due to these two identities
sin(x) = cos(90-x)
cos(x) = sin(90-x)
Why do these identities work? Draw out a right triangle

At this point, it should be fairly clear why those identities work.
Let me know if you are stuck on this part.

So we'll go from
y = 2.5*sin(8x)-1.5
to
y = 2.5*cos(90-8x)-1.5

Graph
https://www.desmos.com/calculator/lobh6fe5te

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