What are the key differences between the graphs of $y = \sin(x)$ and $y = \cos(x)$ ?

$\sin(x)$ : Starts at (0,0). | $\cos(x)$ : Starts at (0,1).

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What are the key differences between the graphs of $y = \sin(x)$ and $y = \cos(x)$ ?

$\sin(x)$ : Starts at (0,0). | $\cos(x)$ : Starts at (0,1).

Compare the symmetry of sine and cosine functions.

Sine: Odd function, symmetric about the origin. | Cosine: Even function, symmetric about the y-axis.

Compare the x-intercepts of $y = \sin(x)$ and $y = \cos(x)$ in the interval $[0, 2\pi]$ .

$\sin(x)$ : 0, $\pi$ , $2\pi$ | $\cos(x)$ : $\frac{\pi}{2}$ , $\frac{3\pi}{2}$

Compare the maximum values of $y = \sin(x)$ and $y = \cos(x)$ .

$\sin(x)$ : Maximum value of 1 at $\frac{\pi}{2}$ | $\cos(x)$ : Maximum value of 1 at 0 and $2\pi$

Compare the minimum values of $y = \sin(x)$ and $y = \cos(x)$ .

$\sin(x)$ : Minimum value of -1 at $\frac{3\pi}{2}$ | $\cos(x)$ : Minimum value of -1 at $\pi$

Compare the effect of a positive phase shift on $\sin(x)$ and $\cos(x)$ .

Both shift the graph to the right by the amount of the phase shift. | The overall shape remains the same, just translated.

Compare the effect of changing the amplitude of $\sin(x)$ and $\cos(x)$ .

Both stretch or compress the graph vertically. | A larger amplitude makes the peaks and troughs more extreme.

Compare the effect of changing the period of $\sin(x)$ and $\cos(x)$ .

Both compress or stretch the graph horizontally. | A smaller period means more cycles within the same interval.

Compare the effect of a vertical shift on $\sin(x)$ and $\cos(x)$ .

Both move the entire graph up or down by the shift amount. | The midline of the graph changes accordingly.

Compare the relationship between sine and cosine to the unit circle.

Sine: y-coordinate on the unit circle. | Cosine: x-coordinate on the unit circle.

What is the general form of a sine function?

$f(x) = A\sin(Bx - C) + D$ , where A is amplitude, B affects the period, C is the phase shift, and D is the vertical shift.

What is the general form of a cosine function?

$f(x) = A\cos(Bx - C) + D$ , where A is amplitude, B affects the period, C is the phase shift, and D is the vertical shift.

How do you calculate the period of a sine or cosine function given 'B'?

Period = $\frac{2\pi}{|B|}$

How is the phase shift calculated in the general form $f(x) = A\sin(Bx - C) + D$ ?

Phase Shift = $\frac{C}{B}$

What is the sine of 0?

$\sin(0) = 0$

What is the cosine of 0?

$\cos(0) = 1$

What is the sine of $\frac{\pi}{2}$ ?

$\sin(\frac{\pi}{2}) = 1$

What is the cosine of $\frac{\pi}{2}$ ?

$\cos(\frac{\pi}{2}) = 0$

What is the sine of $\pi$ ?

$\sin(\pi) = 0$

What is the cosine of $\pi$ ?

$\cos(\pi) = -1$

How do you find the maximum value of $f(x) = A\sin(x) + D$ ?

The maximum value is $A + D$ .

How do you find the minimum value of $f(x) = A\cos(x) + D$ ?

The minimum value is $-|A| + D$ .

How do you determine the period of $f(x) = \sin(Bx)$ ?

Period = $\frac{2\pi}{|B|}$

How do you determine the period of $f(x) = \cos(Bx)$ ?

Period = $\frac{2\pi}{|B|}$

How do you find the x-intercepts of $y = \sin(x)$ in the interval $[0, 2\pi]$ ?

Set $\sin(x) = 0$ and solve for x. The solutions are $x = 0, \pi, 2\pi$ .

How do you find the x-intercepts of $y = \cos(x)$ in the interval $[0, 2\pi]$ ?

Set $\cos(x) = 0$ and solve for x. The solutions are $x = \frac{\pi}{2}, \frac{3\pi}{2}$ .

How do you graph $y = A\sin(x)$ ?

Determine the amplitude (A). 2. Identify key points (0, 0), $(\frac{\pi}{2}, A)$ , $(\pi, 0)$ , $(\frac{3\pi}{2}, -A)$ , $(2\pi, 0)$ . 3. Sketch the curve.

How do you graph $y = A\cos(x)$ ?

Determine the amplitude (A). 2. Identify key points (0, A), $(\frac{\pi}{2}, 0)$ , $(\pi, -A)$ , $(\frac{3\pi}{2}, 0)$ , $(2\pi, A)$ . 3. Sketch the curve.

How do you find the phase shift of $y = \sin(x - c)$ ?

The phase shift is 'c'. If c is positive, the shift is to the right; if c is negative, the shift is to the left.

How do you find the phase shift of $y = \cos(x - c)$ ?