All values of theta are positive. From left to right, the first peak occurs at (pi/2,1) and the second peak occurs at (9pi/2).
A. Write the graphs equation as a cosine function.
B. Write the graphs equation as a sine function.
the period is 4?, so
f(x) = cos(x/2)
since cos(u)=1 at u=0, we have
f(x) = cos((x-?/2)/2)
see
http://www.wolframalpha.com/input/?i=cos((x-%CF%80%2F2)%2F2)+for+0+%3C%3D+x+%3C%3D+5pi
Now recall that cos(x) = sin(?/2 - u)
To write the equation of the graph as a cosine function, we need to consider the amplitude, period, and phase shift.
From the given information, we know that the graph starts at its peak at (pi/2, 1), so the midline is y = 0. The amplitude is the distance between the midline and the peak, so it is 1.
The period of the graph can be found by calculating the difference between the x-coordinates of two consecutive peaks. We see that the second peak occurs at 9pi/2, which is a difference of 8pi from the first peak at pi/2. Since the period is the distance it takes for the graph to complete one cycle, we can conclude that the period is 8pi.
Now, for a cosine function, the general form of the equation is:
y = A * cos(B(theta - C)) + D
Where:
A is the amplitude
B = 2pi / period is the coefficient that determines the frequency (number of cycles) of the graph
C is the phase shift, i.e., the horizontal shift of the graph
D is the vertical shift (also known as the midline)
Using the given information, we have:
A = 1 (amplitude)
B = 2pi / (8pi) = 1/4 (frequency)
C = 0 (no phase shift)
D = 0 (no vertical shift)
Plugging these values into the equation, we get the cosine function in the form:
y = 1 * cos((1/4)(theta - 0)) + 0
Simplifying further, we have:
y = cos((1/4)theta)
Therefore, the equation of the graph as a cosine function is y = cos((1/4)theta).
To write the equation of the graph as a sine function, we can leverage the relationship between cosine and sine functions. Since cosine is just a phase-shifted sine function, we can write the equation of the graph as a sine function with an appropriate phase shift.
The equation for a sine function is similar to the cosine function:
y = A * sin(B(theta - C)) + D
In this case, the phase shift will be pi/2 since the sine graph reaches its peak at (pi/2, 1) while the cosine graph reaches its peak at (0, 1).
Using the same values as before:
A = 1 (amplitude)
B = 2pi / (8pi) = 1/4 (frequency)
C = pi/2 (phase shift)
D = 0 (no vertical shift)
Plugging in the values, we get the sine function in the form:
y = 1 * sin((1/4)(theta - pi/2)) + 0
Simplifying further, we have:
y = sin((1/4)(theta - pi/2))
Therefore, the equation of the graph as a sine function is y = sin((1/4)(theta - pi/2)).