A sinusoidal function has an amplitude of 2 units, a period of 180^circ , and a maximum at (0,3) Represent the function with an equation in two different ways.

## Step 1: Determine the general form of the sinusoidal function<br />### Since the function has a maximum at $(0,3)$, a cosine function with no phase shift is the simplest choice. The general form is $f(x) = A\cos(Bx) + D$.<br /><br />## Step 2: Determine the amplitude $A$<br />### The amplitude is given as 2, so $A=2$.<br /><br />## Step 3: Determine the vertical shift $D$<br />### The maximum value is 3, and the amplitude is 2. The midline is $D = 3 - 2 = 1$.<br /><br />## Step 4: Determine the value of $B$<br />### The period is $180^{\circ}$ or $\pi$ radians. The period is related to $B$ by Period $= \frac{2\pi}{|B|}$. Thus, $\pi = \frac{2\pi}{B}$, which gives $B=2$.<br /><br />## Step 5: Write the first equation<br />### Combining the values of $A$, $B$, and $D$, the first equation is $f(x) = 2\cos(2x) + 1$.<br /><br />## Step 6: Write the second equation using a sine function<br />### A cosine function can be represented as a sine function with a phase shift. Since $\cos(x) = \sin(x + \frac{\pi}{2})$, we can write the second equation as $f(x) = 2\sin(2x + \frac{\pi}{2}) + 1$.