A sinusoidal function has an amplitude of

4 units, a period of 90°, and a maximum at
(0, 2).
a) Represent the function with an equation
using a cosine function.
b) Represent the function with an equation
using a sine function.
How would I get the d and c values?
(y=asin(k(x-d))+c, y=acos(k(x-d))+c

a = 4

k = 360°/90° = 4

so far we would have
y = 4cos (4x), where x would have to be in degrees
our first max would be (0,4) but we want it to be (0,2) , so we have drop our
curve 2 units.
y = 4cos(4x) - 2
this meets all your conditions so don't need a horizontal phase shift, the d

a) y = 4cos (4x) - 2

b) same a, and k
y = 4 sin (4x) -2 , this has a max at (22.5°,2) , but we need it at (0,2)
so we have to shift our curve to the left by 22.5°

y = 4sin (4(x + 22.5°)) - 2

formal way:
we need (0,2) to be on the curve y = 4sin(4(x - d)) - 2
2 = 4sin(4(0-d) - 2
4 = 4sin(-4d)
1 = sin(-4d)
but I know that sin 90° = 1, or take sineinverse(1) to get 90
so -4d = 90
d = -22.5
making our equation:
y = 4 sin (4(x+22.5°) - 2

normally these type of equations would be expressed in radians,
yours would be
a) y = 4cos(4 x) - 2 , to get the k value: 2π/k = π/2 ---> k = 4
b) y = 4sin (4(x + π/8)) - 2

www.wolframalpha.com/input/?i=y+%3D+4cos%284+x%29+-+2+%2C+y+%3D+4sin+%284%28x+%2B+%CF%80%2F8%29%29+-+2

notice the two graphs coincide, and all your stated conditions are met

Well, isn't this a sinusoidal puzzle? Let's have some fun with it!

a) To find the equation using a cosine function, we know the maximum of the function occurs at (0, 2), which means the value of c is 2. The amplitude is 4 units, so the value of |a| is 4. Since cosine functions start at the maximum, there is no phase shift or d value. Finally, the period is given as 90°, which means the value of k is 360°/90° = 4.

Putting it all together, the equation using a cosine function is y = 4cos(4x) + 2. Ta-da!

b) Now, let's represent the function using a sine function. The equation form we have is y = asin(k(x-d)) + c. The maximum of the function still occurs at (0, 2), which means the value of c remains 2. The amplitude is 4 units, so the value of |a| remains 4. However, since sine functions start at zero and then go upward, there is a phase shift or d value of 90°.

So, the equation using a sine function is y = 4sin(4x - 90°) + 2. Voilà!

Remember, these equations involve a fair bit of humor too, so feel free to have a laugh while solving them!

To represent the sinusoidal function using a cosine function, you can follow these steps:

a) The equation for a cosine function is y = A*cos(k(x-d))+c, where A is the amplitude, k is the period, d is the phase shift, and c is the vertical shift.

Given:
Amplitude (A) = 4 units
Period (k) = 90°

To find the values of d and c, we can use the maximum point (0, 2).

We know that the maximum point occurs at the phase shift (d). Since the maximum point is at (0, 2), d = 0.

Also, we know that the maximum value equals the amplitude plus the vertical shift. In this case, the maximum value is 4 units (amplitude) + c. Since the maximum value is 2 units, we can set up the equation: 4 + c = 2.

Solving this equation, we find c = -2.

Therefore, the equation for the function using a cosine function is y = 4*cos(90°x-0°)-2.

b) To represent the function using a sine function, we can use the same steps as above, with the equation y = A*sin(k(x-d))+c.

Using the same given values and the maximum point (0, 2), we find that d = 0 and c = -2.

Therefore, the equation for the function using a sine function is y = 4*sin(90°x-0°)-2.

To represent the given sinusoidal function with an equation using a cosine function, we can use the general cosine function equation:

y = A * cos(B(x - D)) + C

where:
A represents the amplitude
B represents the frequency or angular velocity (2π / period)
D represents the phase shift (horizontal shift)
C represents the vertical shift

a) To determine the values of A, B, D, and C for the given function, we can use the information provided:

Amplitude = 4 units (A = 4)
Period = 90° (B = 2π / 90° = π / 45°)
Maximum at (0, 2), which means the phase shift is 0 since the maximum occurs when x = 0. (D = 0)
The maximum value is 2 units, which is the vertical shift from the x-axis. (C = 2)

Therefore, the equation of the function using a cosine function is:

y = 4 * cos(π/45 * x) + 2

b) To represent the function with an equation using a sine function, we need to use a phase shift of -90° because the given information represents a cosine function, and the sine function is a quarter cycle ahead of the cosine function.

Therefore, the equation of the function using a sine function is:

y = 4 * sin(π/45 * (x - 90°)) + 2

To summarize, by understanding the properties of sinusoidal functions and using the provided information, you can determine the values of A, B, D, and C to represent the function using either a cosine or sine function.