Graphing Trigonometric Functions
Summary
TLDRProfessor Dave's video script offers an in-depth exploration of trigonometric functions, focusing on domain and range, periodicity, and graphing techniques. He explains sine and cosine functions' domains as all real numbers and ranges from -1 to 1. Tangent's domain excludes points where cosine is zero, resulting in an undefined value. The script then delves into graphing transformations, such as amplitude adjustments, horizontal stretches, and phase shifts, using examples like 'Y = 4sin(2x - 2π/3)'. Lastly, it briefly covers the graphs of other trig functions like tangent, cotangent, secant, and cosecant, highlighting their unique characteristics and transformations.
Takeaways
- 📝 The domain of sine and cosine functions is all real numbers, as any angle can be input into these functions.
- 📝 The range of sine and cosine functions is from -1 to 1, inclusive, reflecting the y-values on the unit circle.
- 📝 Tangent function has a domain of all real numbers except where cosine is zero (i.e., not at π/2 + kπ where k is an integer), and its range is all real numbers due to its behavior as cosine approaches zero.
- 📝 Trigonometric functions are periodic with a period of 2π radians, meaning their values repeat every 2π radians.
- 📝 The graph of the sine function starts at 0, reaches 1 at π/2, and returns to 0 at π, then follows a similar pattern in the negative direction.
- 📝 Cosine function is similar to sine but starts at 1 and decreases to 0 at π/2, then to -1 at π, and repeats.
- 📝 Transformations such as amplitude changes, horizontal stretches or compressions, and phase shifts can be applied to trigonometric functions.
- 📝 The amplitude of a trigonometric function is determined by the absolute value of the coefficient in front of the sine or cosine function.
- 📝 The period of a trigonometric function is affected by the coefficient of the variable inside the function, being 2π divided by that coefficient.
- 📝 Vertical shifts move the graph up or down, while horizontal shifts (phase shifts) affect the starting point of the function, influenced by the coefficient of the variable.
- 📝 Other trigonometric functions like tangent, cotangent, secant, and cosecant have different graphs with asymptotes where the original functions had zeros.
Q & A
What are the domain and range of the sine function?
-The domain of the sine function is all real numbers, as it can accept any angle. The range is from negative one to one, inclusive, as the Y values on the unit circle fall within this interval.
How does the domain of the cosine function compare to that of the sine function?
-The domain of the cosine function is the same as that of the sine function, which is all real numbers, because it can also accept any angle.
Why is the range of the tangent function different from that of the sine and cosine functions?
-The range of the tangent function is all real numbers because as cosine approaches zero, the value of tangent (which is sine over cosine) approaches infinity.
What are the restrictions on the domain of the tangent function?
-The domain of the tangent function is almost all real numbers, but it cannot be evaluated when cosine is zero, which occurs at angles of half pi plus or minus multiples of pi.
What is the period of the sine and cosine functions?
-The period of the sine and cosine functions is two pi radians, as their values repeat every two pi radians.
How can you graph the sine function over one period?
-You start by plotting points on the Y-axis corresponding to the sine values at multiples of pi/6, then continue through the unit circle, noting the sine values decrease to zero at pi, become negative in the third quadrant, and return to zero at two pi.
What transformations can be applied to the sine function?
-Transformations include applying a coefficient to stretch or shrink the function (changing amplitude), a coefficient on X for horizontal stretch or compression (changing period), and vertical or horizontal shifts.
How does the amplitude of the sine function relate to the coefficient in front of it?
-The amplitude of the sine function is equal to the absolute value of the coefficient in front of it, such as in Y = A * sin(X).
What is the effect of a coefficient operating on X in the sine function?
-A coefficient operating on X in the sine function causes a horizontal stretch or compression, changing the period. For example, in Y = sin(B * X), the period is two pi over B.
How do vertical shifts affect the graph of the sine function?
-Vertical shifts add or subtract a constant from the sine function, moving the graph up or down without altering its shape. For instance, Y = sin(X) + 1 shifts the graph up by one unit.
What is the relationship between the graphs of the sine and cosine functions?
-The graphs of the sine and cosine functions are similar, with the cosine function being a phase shift of pi/2 to the right of the sine function.
Outlines
📐 Introduction to Trigonometric Functions
Professor Dave introduces the concept of graphing trigonometric functions, starting with the domain and range of sine and cosine. He explains that the domain of sine and cosine is all real numbers, as any angle can be plugged into these functions. Their range is between -1 and 1, inclusive, as seen on the unit circle. Tangent, being sine over cosine, has a domain that excludes angles where cosine is zero (i.e., multiples of π/2), and its range is all real numbers since it approaches infinity as cosine nears zero. The paragraph also discusses the periodic nature of these functions, with a period of 2π radians, and how their values repeat after each period.
🔍 Graphing Sine and Cosine Functions
The script describes the process of graphing the sine and cosine functions, emphasizing their periodic nature and how they repeat every 2π radians. It details how to plot the sine function over one period, showing how its values change from 0 to 1 and back to 0 as X increases from 0 to π/2, and then decreases to -1 and back to 0 as X goes from π/2 to π. The transformations that can be applied to these functions, such as amplitude changes, horizontal stretches, and vertical or horizontal shifts, are also explained. An example is given for graphing a transformed sine function, Y = 4sin(2X - 2π/3), where the amplitude, period, and phase shift are calculated.
📉 Exploring Other Trigonometric Graphs
The final paragraph discusses the graphs of the other trigonometric functions, such as tangent, cotangent, cosecant, and secant. It highlights the differences in their appearance compared to sine and cosine, particularly the vertical asymptotes where the functions are undefined. The tangent function is noted for its infinite approach as cosine nears zero and its undefined nature at these points. Cotangent is described as the reciprocal of tangent, with a similar graph but inverted and shifted. Cosecant and secant also have asymptotes at points where sine and cosine are zero. The paragraph concludes with a note on the expected transformations for these functions, mirroring those of sine and cosine.
Mindmap
Keywords
💡Trigonometric functions
💡Unit circle
💡Domain
💡Range
💡Periodic
💡Transformations
💡Amplitude
💡Phase shift
💡Vertical shift
💡Asymptotes
💡Cosecant, Secant, Cotangent
Highlights
Introduction to graphing trigonometric functions
Domain and range of sine and cosine functions
Domain of sine and cosine is all real numbers
Range of sine and cosine is from negative one to one
Domain and range of tangent function
Tangent function approaches infinity as cosine approaches zero
Periodic nature of trigonometric functions
Period of sine and cosine is two pi radians
Graphing sine function over one period
Graphing cosine function and its similarity to sine
Transformations of trigonometric functions
Amplitude affects the range of sine function
Coefficient on X causes horizontal stretch or compression
Vertical shifts in trigonometric functions
Graphing transformed function: four sine of two X minus two thirds pi
Finding amplitude, period, and phase shift for transformed functions
Graphs of other trigonometric functions: tangent, cotangent, secant, and cosecant
Tangent function has vertical asymptotes where cosine is zero
Transformations apply to all trigonometric functions
Comprehension check on graphing and transformations of trigonometric functions
Transcripts
Professor Dave here, let’s graph some trig functions.
We’ve just spent some time learning about the six trigonometric functions, as well as
how they relate to one another, and the unit circle.
Just like any other function, we will want to be able to graph them, but there are some
interesting things that we can point out about these functions first.
First, let’s talk about the domain and range of the functions.
Looking back at the unit circle, we know that each point has the coordinates cosine theta,
sine theta.
So looking first at sine theta, the domain of this function is the set of all the angles
we could plug into this function, and the range of the function is all the possible
values for sine theta.
This means that the domain of sine theta is all real numbers, since we could plug in any angle.
It could be greater than two pi radians, we just end up going around and around indefinitely
as we approach infinity.
And we could also have negative angles, we just go the other way.
But the range of the function is negative one to one, inclusive.
Sine theta can only have values within this interval, as we can clearly see from the unit
circle, as the Y values of all of these points are somewhere in between negative one and one.
The same goes for cosine theta, we could plug in any angle, but if we look at the X values
of these points, they always fall between negative one and one.
So both sine and cosine will have a domain of all real numbers and a range of negative
one to one.
Tangent is different, because that’s sin over cosine, and as cosine gets very close
to zero, in either direction, the function gets very big, approaching infinity.
So the range of tangent theta will be all real numbers.
The domain is almost all real numbers, but we can’t evaluate tangent when cosine is
zero, because anything over zero is undefined.
Cosine is zero at half pi and three halves pi, so the domain of tangent theta is all
real numbers except half pi plus or minus multiples of pi.
The other thing we want to understand about these trig functions is that they are periodic.
Their values repeat, over and over again, as we go through each period of the function.
The period of these functions is two pi radians, because after two pi radians, we are back
to where we started and all the values repeat.
In other words, the sin of X plus two pi is the same as the sine of X.
The same goes for cosine.
Let’s go ahead and graph the sine and cosine functions now, so we can see exactly what
they look like.
Let’s bring up the coordinate plane, and also the unit circle.
Looking at the Y coordinates of these points, we can start to plot the graph of Y equals
sine X, going in multiples of pi over six.
When X is zero, Y is zero.
When X is pi over six, Y is one half.
When X is pi over three, Y is root three over two.
When X is half pi, Y is one.
Then as we move through quadrant two of the unit circle, the sine values start to go back
down, until we get to X equals pi, where Y is again zero.
Now as we enter quadrant three, we start to get negative values for the Y coordinate,
and thus negative values for sine X.
These will be the same values as the first two quadrants, just negative, until we get
to negative one.
And then moving through the fourth quadrant, we get back to zero.
So there is the graph of Y equals sine X for one period of the function.
Once we get to two pi radians, it’s the same as being at zero, and when we enter another
period of the function, all of the values will repeat.
That’s what makes this a periodic function.
If we graph multiple periods of the function, it looks like this, and we can clearly see
its cyclical nature.
How can we manipulate this function?
Well, we can apply any of the transformations we learned in algebra.
If we put a coefficient here, that will stretch or shrink the function, which will change
the amplitude of the function.
Y equals two sine X will look the same, except that all the values are doubled, so we are
cycling between two and negative two.
If the coefficient is negative, it reflects everything across the X axis, which looks
like this.
In this way, for any function in the form Y equals A sine X, the amplitude will be equal
to the absolute value of A. If instead, there is a coefficient operating on X, that would
be a horizontal stretch, as Y equals sine of two X would mean that the function will
rise and fall at twice the normal rate.
This would mean that the period of this function is pi instead of two pi, and in fact, for
any function in the form of Y equals A sine BX, the period is two pi over B. Lastly, we
can have vertical or horizontal shifts.
If we have a term that is being added to the function, that’s a vertical shift, just
like we saw with parabolas.
Y equals sin X plus one will just shift everything up one.
If instead we have some number inside the function, like Y equals sine of the quantity
X plus half pi, this whole thing will shift half pi to the left.
As it happens, we have just generated the graph for Y equals cosine X.
This will make sense if we refer to the unit circle again, and see that the cosine of zero
is one.
That means the function must start up here.
Then the cosine decreases until we get to zero at half pi, then goes towards negtive
one at pi.
Then it’s back to zero at three-halves pi, and then up to one at two pi, after which
things repeat.
So the graphs for sine and cosine are extremely similar, they are just slightly shifted, and
all of the transformations we used for sine X will apply for cosine as well.
Let’s use what we’ve just learned to graph one period of four sine of the quantity two
X minus two thirds pi.
First let’s find the amplitude.
We can get that from this number here, which is four, so the function will move between
negative four and four.
Then, let’s find the period.
That will be two pi over this term, so the period will be equal to pi.
Then we find the phase shift.
This term would mean that the whole thing is shifted two thirds pi to the right from
the origin, but we have to divide that by this number, since this is causing things
to contract, so that leaves us with pi over three.
So the period will start at one third pi, and end at four thirds pi, going up to four,
and then down to negative four, before coming back to zero.
We can find the X coordinates of these key points as well, we just have to split the
period up into four parts.
That gives us a quarter pi, and we just add quarter pi successively, starting with a third
pi, to get all the X coordinates.
To do that, we need a common denominator, which will be twelve, and then we combine
and reduce, using the rules we already learned for adding fractions.
Before we move on from this subject, let’s just quickly look at the graphs of the other
trig functions so that we know what they look like.
Tangent looks quite a bit different, because the range is no longer negative one to one,
like for sine and cosine.
Instead, this function approaches positive and negative infinity as cosine values approach
zero, and when cosine is zero, tangent is undefined, so we have a series of vertical
asymptotes.
Here is the graph for Y equals tangent X.
It has a period of pi, because tangent values from quadrant one, where sine and cosine are
both positive, come back again in quadrant three, where sine and cosine are both negative.
Likewise, tangent values in quadrant two, which are positive over negative, come back
in quadrant four, where it’s negative over positive.
This function can be transformed in all the same ways that we saw for sine and cosine.
Cotangent will be similar, but as it is the reciprocal of tangent, it’s a little different,
falling to the right instead of rising, and shifted slightly, since the X values that
originally generated asymptotes will now generate Y values of zero, as one over infinity is
zero, and the points that originally had Y values of zero, will now contain asymptotes,
as one over zero is undefined.
Cosecant and secant will also look a little funny, as all the points where the sine and
cosine equal zero will now be asymptotes for their reciprocal functions, again because
one over zero is undefined.
The transformations for all these graphs are exactly what you would expect, so having gone
over this already, let’s check comprehension.
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