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.

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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 $\pi$ ?

$\cos(\pi) = -1$

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.

Define 'sine' in the context of the unit circle.

The y-coordinate of a point on the unit circle corresponding to a given angle.

Define 'cosine' in the context of the unit circle.

The x-coordinate of a point on the unit circle corresponding to a given angle.

What is the period of the sine function?

The interval over which the sine function completes one full cycle, which is $2\pi$ .

What is the period of the cosine function?

The interval over which the cosine function completes one full cycle, which is $2\pi$ .

Define the term 'amplitude' in the context of sine and cosine functions.

The amplitude is the maximum displacement of the function from its midline. For standard sine and cosine, it's 1.

What is the range of the sine function?

The set of all possible output values of the sine function, which is [-1, 1].

What is the range of the cosine function?

The set of all possible output values of the cosine function, which is [-1, 1].

Define 'x-intercept' in the context of sine and cosine graphs.

The points where the graph intersects the x-axis, i.e., where the function value is zero.

What is a 'vertical shift' in the context of trigonometric functions?

A transformation that moves the entire graph up or down by a constant value.

Define 'phase shift' in the context of sine and cosine functions.

A horizontal shift of the sine or cosine graph, often represented as $sin(x - c)$ or $cos(x - c)$ , where 'c' is the phase shift.