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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.
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 $y = A\sin(x)$?
1. 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 $y = A\cos(x)$?
1. 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 $y = \cos(x - c)$?
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.
Explain how the unit circle relates to the sine function's graph.
The y-coordinates of points on the unit circle, as you move counterclockwise around the circle, correspond to the values of the sine function at those angles.
Explain how the unit circle relates to the cosine function's graph.
The x-coordinates of points on the unit circle, as you move counterclockwise around the circle, correspond to the values of the cosine function at those angles.
Describe the key features of the sine graph.
Starts at (0,0), oscillates between -1 and 1, has a period of $2\pi$, and is an odd function (symmetric about the origin).
Describe the key features of the cosine graph.
Starts at (0,1), oscillates between -1 and 1, has a period of $2\pi$, and is an even function (symmetric about the y-axis).
Explain the effect of changing the amplitude of a sine or cosine function.
Changing the amplitude stretches or compresses the graph vertically. A larger amplitude means a greater maximum and minimum value.
Explain the effect of changing the period of a sine or cosine function.
Changing the period compresses or stretches the graph horizontally. A smaller period means the function oscillates more frequently.
Explain the effect of a vertical shift on a sine or cosine function.
A vertical shift moves the entire graph up or down. Adding a constant 'D' to the function shifts it up by 'D' units.
Explain the effect of a phase shift on a sine or cosine function.
A phase shift moves the entire graph left or right. It's a horizontal translation determined by the value C/B in the general form.
Why are sine and cosine functions called periodic functions?
Because their graphs repeat their pattern over regular intervals (periods).
How do sine and cosine functions relate to right triangles?
In a right triangle, sine is the ratio of the opposite side to the hypotenuse, and cosine is the ratio of the adjacent side to the hypotenuse.