06Dsincosgraph

toc = Graphs of Sine and Cosine =

The sine curve has many applications in Physics. For example, sound, light and electromagrnetic waves all travel in a sine wave.

For all trig graphs,
 * the ** median line ** is a horizontal line through the centre of the graph
 * the ** amplitude ** is the maximum height above the median line
 * the ** period ** is the distance in the x-direction to complete one cycle of the graph

Graph of y = sin(x)
The graph of y = sin(x) {sometimes called the ** sine wave **} is derived from the definition of sin using the unit circle.



The above diagram only shows one cycle of sine. The sine function actually continues to infinity.



For the graph, y = sin(x)
 * The median line is along the x-axis (y = 0)
 * The amplitude is 1
 * The period is 2 p
 * Domain: x Î R
 * Range: y Î [–1, 1]

Graph of y = cos(x)
The graph, y = cos(x) is the same shape as y = sin(x) but translated p /2 to the left.

For the graph, y = cos(x)
 * The median line is along the x-axis (y = 0)
 * The amplitude is 1
 * The period is 2 p
 * Domain: x Î R
 * Range: y Î [–1, 1]

Transformations
We can use the standard transformations, such as dilations, translations and reflections on the trig graphs.

Dilations in the Y-Direction: y = asin(x)
y = asin(x) causes a dilation by a factor of **a** in the y-direction, in the same way that y = af(x) causes a dilation.

Because the amplitude of sin(x) is 1, a dilation by a factor of **a** changes the amplitude to **a**.

Recall that if **a** is negative, we get a reflection across the x-axis (in the y-direction).


 * Note: ** Since amplitude > 0, the amplitude is the magnitude of a.

** Example 1 **
For the graph, y = 2sin(x)
 * The median line is along the x-axis (y = 0)
 * Amplitude = 2
 * Period = 2 p
 * Domain: x Î R
 * Range: y Î [–2, 2]

The x-intercepts can be found by solving 2sin(x) = 0 {we use k instead of n because n has a different meaning here}
 * x-intercepts: x = k p, k Î J

Dilations in the X-Direction: y = cos(nx)
y = cos(nx) causes a dilation by a factor of **1/n** in the x-direction, in the same way that y = f(nx) causes a dilation.

Because the period starts at 2 p, a dilation by 1/n changes the period to 2 p /n.

Recall that if n is negative, we get a reflection across the y-axis (in the x-direction).


 * Note: ** Since period > 0, the period is 2 p divided by the magnitude of n.

** Example 2 **
For the graph, y = cos(2x)
 * The median line is along the x-axis (y = 0)
 * Amplitude = 1
 * Period = p
 * Domain: x Î R
 * Range: y Î [–1, 1]

The x-intercepts can be found by solving cos(2x) = 0 math .\quad \centerdot \; \text{ x-intercepts:} \; x=k\pi \pm \dfrac{\pi}{4}, \; k \in Z math or, more simply: math .\quad \centerdot \; \text{ x-intercepts:} \; x= \left( 2k+1 \right) \dfrac{\pi}{4}, \; k \in Z math

** Example 3 **
math \text{For the graph } y=\sin \left( x-\dfrac{\pi}{2} \right)+2 math
 * Median line: y = 2
 * Amplitude = 1
 * Period = 2 p
 * Domain: x Î R
 * Range: y Î [1, 3]

The points on the median line can be found by solving y = 2. math .\quad \centerdot \; \text{ median points: } x = \left( 2k+1 \right) \dfrac{\pi}{2}, \; k \in Z math

x-intercepts (if there were any) could have been found by solving y = 0.

Summary
y = asin(n(x – b)) + c {and same for cos}
 * dilated by a factor of a in the y-direction
 * dilated by a factor of 1/n in the x-direction
 * if a < 0, reflected in the y-direction
 * if n < 0, reflected in the x-direction
 * translated b units to the right
 * translated c units up

SO: Go to top of page flat
 * Median line: y = c
 * Amplitude = |a|
 * Period = 2 p /n
 * Domain: x Î R
 * Range: y Î [c + a, c – a]
 * Median points: solve y = c
 * x-intercepts: solve y = 0

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