a) Determine the equation of a sine function that would have a range of {y€R|-7<(or equal) Y <(or equal) 1} and a period of 135degrees.
b) determine the cosine function that results in the same function an part (a).
Any help provided will be greatly appreciated.
your range is 8, so you need
4cos ??
period = 360/k
360/k = 135
135k = 360
k = 8/3
so far we have
y = 4cos (8/3)Ø
that would give us a max value of 4, but we want our max to be 1, so we have to drop our curve by 3 units
final:
y = 4cos(8/3)Ø - 3
Ok I was wondering why the zero with the slash was in the equation, i have never seen that before!
To determine the equation of a sine function with a given range and period, follow these steps:
a) Given the range {y€R |-7≤y≤1} and the period of 135 degrees:
1. Determine the amplitude: The range is between -7 and 1, so the amplitude is half the difference between the maximum and minimum values: Amplitude = (1 - (-7))/2 = 8/2 = 4.
2. Determine the vertical shift (midline): Since the range is between -7 and 1, the midline is halfway between them: Midline = (-7 + 1)/2 = -6/2 = -3.
3. Determine the phase shift: Since the period is given in degrees, convert it to radians by multiplying by π/180: Phase shift = 135 * (π/180) = 3π/4.
4. Determine the equation: The general equation for a sine function is y = A*sin(B(x - C)) + D, where A is the amplitude, B is the frequency (or 2π divided by the period), C is the phase shift, and D is the vertical shift.
Putting it all together, the equation for the sine function is:
y = 4*sin((2π/135)(x - 3π/4)) - 3.
b) To determine the cosine function that results in the same function as in part (a), you can use the phase shift property of cosine.
The phase shift property states that the cosine function with a phase shift of θ is equivalent to the sine function with a phase shift of θ - π/2.
So, the equation for the cosine function is:
y = 4*cos((2π/135)(x - (3π/4 - π/2))) - 3.
Simplifying further:
y = 4*cos((2π/135)(x - 7π/4)) - 3.
These equations will result in a sine and cosine function with the given range and period.