SOLUTION: If we change the circle from the standard unit circle to the circle with equation (x-2)^2+(y+5)^2=9 and we change the point Q(1,0) to be Q(4,√5-5). Define P(x,y) as before an

Question 1079474: If we change the circle from the standard unit circle to the circle with equation (x-2)^2+(y+5)^2=9 and we change the point Q(1,0) to be Q(4,√5-5). Define P(x,y) as before and let f(t)=x and g(t)=y.
a. Find equations for f(t) and g(t) as transformations of cos t and sin t.
b. Explain the significance of each of the attributes of the sinusoid functions in the previous part.
Confused. Can someone help me with this?
Thanks
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Is that the whole story? Or did we just walk in in the middle of the movie?
Is t an angle defined in some way?
It sound as if t is time, and you have a point P(x,y)
that started at some position
(maybe all the way to the right),
and is moving around the circle (maybe counterclockwise)
at some angular velocity (say radians per second).
With all those assumptions (and t in seconds),
a point moving around the unit circle
would start at Q(1,0), and t seconds later its coordinates would be
and .
With some luck w=1, and it is just cos(t) and sin(t).
The new circle has center (2,-5) and radius .
To make matters worse, the point would instead start
2 units to the right and units up from the center,
at

.
That point has already moved an angle ,
with
from the expected starting point,
which was all the way to the right of the circle center.
We have to add to the angle for sine and cosine,
and we have to account for the larger (3) radius.
After t seconds that point will be
to the right, and
up from the circle center, at
and (or , if you do not want to start with a minus sign).
Maybe that would cover part a.

As for part b, I do not know what "attributes" were meant.
Different classes, different jargon, emphasis on different concepts/ideas.
Pick the words that resonate with your class experience.
The factor 3 is the of the sinusoidal functions.
It was changed from what you had in the unit circle,
"stretching" or "expanding" or "dilating" the functions vertically.
the added is a phase shift of the wave,
or horizontal translation of the function.
The 2 and -5 added to the functions for x and y
are vertical translations of the functions.
The period is

EDIT:
If P has moved a distance counterclockwise around a circle of radius , centered at O(0,0),
the ray OP has swept an angle measuring radians.
(In higher math we measure angles in radians, as if degrees are for elementary school only).
By convention, we consider counterclockwise sweeps as positive angles, clockwise sweeps as negative angles.
That way, we can say "do a turn", meaning to turn right at the corner,
or "give the knob a turn," meaning to turn it turns counterclockwise.
Then we define the trigonometric functions and
based on the coordinates of P.

If we use a circle with radius and center C(2,-5),
for a point P(x,y) moving a distance counterlockwise,
the ray CP would sweep an angle measuring radians.
Point

is
units to the right, and
units up from C(2,-5),
at units distance from C, measured along a slanted line.
That line forms an angle theta with the horizontal, with , which corresponds to about , in radians.
Ray CQ would be a counterclockwise sweep from (5,-5),
the point in the circle directly to the right of C.

We can write expressions for the new functions,
based on geometry:
With respect to C, Q would be
units to the right, and
units up.
Its coordinates would be
, and ,
so and .
Those functions are sinusoidal functions with amplitude , and period . would be considered a phase shift.

You could also look at the new functions as transformations of the "parent" functions: cos(t) and sin(t) .
With respect to their parent functions,
f(t) and g(t) were
first "dilated" by a factor of vertically and horizontally, because the circle was expanded by that factor.
There were 2 more changes after that, and I do not think the order matters.
The circle center was shifted 2 units right and 5 down,
shifting 2 units up, and shifting 5 units down.
Also the starting point was changed from (1,0) to Q,
shifting both functions to the right, because P was given a running start.
Those transformations are effected by the factors (dilations) , or added terms (shifts or translations).
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SOLUTION: If we change the circle from the standard unit circle to the circle with equation (x-2)^2+(y+5)^2=9 and we change the point Q(1,0) to be Q(4,&#8730;5-5). Define P(x,y) as before an

SOLUTION: If we change the circle from the standard unit circle to the circle with equation (x-2)^2+(y+5)^2=9 and we change the point Q(1,0) to be Q(4,√5-5). Define P(x,y) as before an