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: Hidden categories:& 2. Graphs of `y = a sin bx` and
`y = a cos bx`
2. Graphs of y = a sin bx and y = a
by M. Bourne
Interactive applet
Don't miss later on this page:
The variable b in both of the following graph types affects the period (or wavelength) of the graph.
y = a sin bx
y = a cos bx
The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again.
Graph Interactive - Period of a Sine Curve
Here's an applet that you can use to explore the concept of period and frequency of a sine curve.
Frequency is defined as `&frequency& = 1/&period&`. We'll see more on this below.
In this applet, a point on a circle rotates at a constant rate, and its height at time `t` traces out a sine curve.
Things to Do
Clik the Start button
At first you'll see a sine curve traced out as the circle rotates.
Now, change the value of b using the slider. If you increase b, the period for each cycle will go down and the frequency will increase. Observe the dot on the circle also goes around more quickly.
Observe the number of cycles that you see between t = 0 and t = 2& (= 6.28). For
b = 1 you see one cycle, for b = 2, you see 2 cycles, and so on.
The period of the curve is marked with a red vertical line.
The units along the horizontal (time) axis are in radians. So & = 3.14 radians and 2& = 6.28 radians.
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Did you notice?
The variable `b` gives the number of cycles between `0` and `2pi`.
Higher `b` gives higher frequency (and lower period).
Formula for Period
The relationship between `b` and the period is given by:
`"Period"=(2pi)/b`
Note: As b gets larger, the period
decreases.
Changing the Period
Now let's look at some still graphs to see what's going on.
The graph of
y = 10 cos x, which we learned about in the last section, , is as follows.
The graph of `y=10cos(x)` for `0 & x & 2pi`
As we learned, the period is `2pi`.
Next we see
y = 10 cos 3x. Note the `3` inside the cosine term.
0.5π
1.5π
The graph of `y=10cos(3x)` for `0 & x & 2pi`
Notice that the period is different. (However, the amplitude is `10` in each example.)
This time the curve starts to repeat itself at `x=(2pi)/3`, which is marked with a red vertical line. This is consistent with the formula we met above, which siad:
`"Period"=(2pi)/b`
Now let's view the 2 curves on the same set of axes. Note
that both graphs have an amplitude of `10` units, but their period
is different.
0.5π
1.5π
`y = 10 cos(x)`
`y = 10 cos(3x)`
The graphs of `y=10cos(x)` and `y=10cos(3x)` for `0 & x & 2pi`
Interactive: Spring with mass
When you stretch (or compress) a spring then let go, it will vibrate back and forward. It will continue to do so if there are no other forces acting on it. (In reality, the spring slows down due to friction and the force of gravity.)
The vibration is periodic, and we can describe it using a sine or cosine curve.
The period of a spring's motion is affected by the stiffness of the spring (usually denoted by the variable k), and the mass on the end of the spring (m). You can investigate this property in the following interactive graph.
Things to do
Observe the curve we get as the spring vibrates. It's a cosine curve.
Observe the period of the cosine curve.
Vary the stiffness of the spring (`k`) and see the effect on the period.
Vary the mass (`m`) and observe the effect on the period.
What is the period when mass = stiffness (`m=k`)? Why?
For information, the period of a vibrating spring with stiffness
k and with mass m on the end, is given by: `T=2 pi sqrt(m/k)`.
The equation of the cosine curve you'll see is `h = h_1 + a cos sqrt(k/m)t`, where `h=` height at time `t`, `a = ` amplitude of the motion, and `h_1` is an offset from the `t`-axis due to the spring stiffness and/or mass of the object.
In this applet, the position of `2pi` is fixed.
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Once again, a real spring would actually slow down as time goes on. Also, if we increase the mass, the spring will stretch out more, and if it's stiffer, it will stretch less. However, to keep things simple, the situation is idealized.
In this example, the spring is actually moving sideways on a table and we are looking from above. (We assume gravity is not a factor.)
Good to Know...
Tip 1: The number b tells us the number of cycles in each
For y = 10 cos x, there is one cycle
between `0` and 2& (because b = 1).
For y = 10 cos 3x, there are 3 cycles
between `0` and 2& (because b = 3).
Tip 2: Remember, we are now operating using RADIANS. Recall that:
2& = 6.283185...
We only use radians in this chapter.
For a reminder, go to:
Need Graph Paper?
1. Sketch 2 cycles of y = 3 cos 8x.
2. Sketch 2 cycles of y = cos 10x.
3. Sketch 2 cycles of y = 5 sin 2&x.
4. Sketch 2 cycles of `y = 4\ sin\ x/3`
Pistons don't exactly follow a sine path
The movement of a piston in an engine is often quoted as an example of a sine curve. It almost is, but only under certain conditions. See
to investigate this further, using an interactive animation.
Defining Sine
Curves using Frequency
It is common in electronics to express the sin graph in terms of the frequency f as follows:
y = sin 2&ft
This is very convenient, since we don't have to do any calculation to find the frequency (like we were doing above). The frequency, f, is normally measured in cycles/second, which is the same as Hertz (Hz).
The period of the curve (the time it takes to go from one crest to the next crest) can be found easily once we know the frequency:
The units for period are normally seconds.
Household voltage in the UK is alternating current, `240\ &V&` with frequency `50\ &Hz&`. What is the equation describing this voltage?
Coming Next
In the next section we learn about phase shift.
Later, we learn some .
But first, let's see another application of frequency.
Music Example
The frequency of a note in music depends on the period of the wave. If the frequency is high, if the frequency is low, the period is longer.
A student recently asked me
an interesting question. She wanted to know the frequencies of all the notes on a piano.
A piano is tuned to A = 440 Hz (cycles/second) and the other
notes are evenly spaced, 12 notes to each octave. A note an
octave higher than A = 440 Hz has twice the frequency (880 Hz) and an
octave lower
than A = 440 Hz has half the frequency (`220\ &Hz&`).
Click here to find out the
Related, useful or interesting IntMath articles
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In trigonometric graphs, is phase angle the same as phase shift? A reader challenges my statement.
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See the Interactive Mathematics .扫二维码下载作业帮
3亿+用户的选择
下载作业帮安装包
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数学解方程（1）8sin²x＝3sin2x-1 （2）（sinx＋cosx）²＝2cos2xkπ＋arctan1/3 2,kπ＋arctan1/3或kπ-π/4,
作业帮用户
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（1）8sin²x＝3sin2x-18sin²x＝6sinxcosx-18sin²x-6sinxcosx+sin²x+cos²x=09sin²x-6sinxcosx+cos²x=0(3sinx-cosx)²=03sinx-cosx=0tanx=sinx/cosx=1/3x=kπ+arctan1/3（2）（sinx＋cosx）²＝2cos2xsin²x+2sinxcosx+cos²x=2-4sin²x4sin²x+2sinxcosx+1=24sin²x+2sinxcosx-1=04sin²x+2sinxcosx-sin²x-cos²x=03sin²x+2sinxcosx-cos²x=0(sinx+1)(3sinx-1)=0sinx=-1、sinx=1/3kπ-π/4或kπ＋arctan1/3
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扫描下载二维码& 1. Graphs of `y = a sin x` and
`y = a cos x`
1. Graphs of
by M. Bourne
(a) The Sine Curve y = a sin t
We see sine curves in many naturally occuring phenomena, like water waves. When waves have more energy, they go up and down more vigorously. We say they have greater amplitude.
Let's investigate the shape of the curve
a sin t and see what the concept of &amplitude& means.
Have a play with the following interactive.
Sine curve Interactive
You can change the circle radius (which changes the amplitude of the sine curve) using the slider.
The scale along the horizontal t-axis (and around the circle) is radians. Remember that & radians is `180&`,
so in the graph, the value of `pi = 3.14` on the t-axis represents `180&` and `2pi = 6.28` is equivalent to `360&`.
y = 70 sin(0) = 0
Amplitude:
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Did you notice?
The shape of the sine curve forms a
regular pattern (the curve repeats after the wheel has gone around once). We say such curves are periodic. The period is the time it takes to go through one complete cycle.
In the interactive, when the radius of the circle was `50` units then the curve went up to `50` units and down to `-50` units on the y-axis. This quantity of a sine curve is called the amplitude of the graph. This indicates how much energy is involved in the quantity being graphed. Higher amplitude means greater energy.
The rotation angle in radians is the same as the time (in seconds). See more on . All the graphs in this chapter deal with angles in radians. Radians are much more useful in engineering and science compared to degrees.
When the angle is in the first and second quadrants, sine is positive, and when the angle is in the 3rd and 4th quadrants, sine is negative.
[Credits: The above animation is loosely based on a demo graph by HumbleSoftware.]
The &a& in the expression y =
x represents the amplitude of the graph. It is an indication of how much energy the wave contains.
The amplitude is the distance from the &resting& position (otherwise known as the mean value or average value) of the curve. In the interactive above, the amplitude can be varied from `10` to `100` units.
Amplitude is always a positive quantity. We could write this using
signs. For the curve y = a sin x,
amplitude `= |a|`
Graph of Sine x - with varying amplitudes
We start with y = sin x.
It has amplitude `= 1` and period `= 2pi`.
The graph of `y=sin(x)` for `0 & x & 2pi`
Now let's look at the graph of y = 5 sin x.
This time we have amplitude = 5 and period = 2&. (I have used a different scale on the y-axis.)
The graph of `y=5sin(x)` for `0 & x & 2pi`
And now for y = 10 sin x.
Amplitude = 10 and period = 2&.
The graph of `y=10sin(x)` for `0 & x & 2pi`
For comparison, and using the same y-axis scale, here are the graphs of
= 5 sin x and
= 10 sin x
on the one set of axes.
Note that the graphs have the same period (which is `2pi`) but different
amplitude.
The graphs of `p(x), q(x)`, and `r(x)` for `0 & x & 2pi`
(b) Graph of Cosine x - with varying amplitudes
Now let's see what the graph of y = a cos x looks like. This time the angle is measured from the positive vertical axis.
Cosine curve Interactive
Similar to the sine interactive at the top of the page, you can change the amplitude using the slider.
Click "Start" to see the animation.
y = 100 cos(0) = 0
Amplitude:
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Did you notice?
The sine and cosine graphs are almost identical, except the cosine curve starts at `y=1` when `t=0` (whereas the sine curve starts at `y=0`). We say the cosine curve is a sine curve which is shifted to the left by `&/2\ (= 1.57 = 90^@)`.
The value of the cosine function is positive in the first and fourth quadrants (remember, for this diagram we are measuring the angle from the vertical axis), and it's negative in the 2nd and 3rd quadrants.
Now let's have a look at the graph of the simplest cosine curve,
y = cos x (= 1 cos x).
The graph of `y=cos(x)` for `0 & x & 2pi`
We note that the
amplitude `= 1` and period `= 2&`.
Similar to what we did with y = sin x above,
we now see the graphs of
p(x) = cos
q(x) = 5 cos
r(x) = 10 cos
on one set of axes, for comparison:
The graphs of `p(x), q(x)`, and `r(x)` for `0 & x & 2pi`
Note: For the cosine curve, just like the sine curve, the period of each graph is the same (`2pi`), but
the amplitude has changed.
Need Graph Paper?
Sketch one cycle of the following without using a
table of values! (The important thing is to know the shape of these
graphs - not that you can join dots!)
Each one has period `2 pi`. We learn more about period in the next section .
The examples use t as the independent variable. In electronics, the variable is most often t.
1) i = sin t
2) v = cos t
Related, useful or interesting IntMath articles
Here's a light-hearted introduction to the concepts of trigonometric graphs.
Gaisma has many interesting day/night graphs, which are (almost) sine curves.
In trigonometric graphs, is phase angle the same as phase shift? A reader challenges my statement.
IntMath forum
Latest Graphs of the Trigonometric Functions forum posts:Got questions about this chapter?
by mfaisal_1981 [Solved!] by Andrew [Solved!] by Rismiya [Solved!] by Joe [Solved!]
Search IntMath, blog and Forum
Online Algebra Solver
This algebra solver can solve a wide range of math problems.
Trigonometry Lessons on DVD
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More info:
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
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See the Interactive Mathematics .