Graphing Trigonometric Functions

Summary

Professor 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

00:00

📐 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.

05:01

🔍 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.

10:06

📉 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.

Keywords

💡Trigonometric functions

Trigonometric functions, also known as trig functions, are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the context of the video, these functions are used to describe the relationship between angles and their corresponding coordinates on the unit circle. The six primary trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. The video focuses on the first three, which are fundamental to understanding periodic functions and their graphical representations.

💡Unit circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It plays a crucial role in defining trigonometric functions. In the video, the unit circle is used to explain how each point on the circle corresponds to an angle (in radians) and has coordinates that represent the values of cosine and sine for that angle. This visual representation helps in understanding the periodic nature and range of these functions.

💡Domain

The domain of a function refers to the set of all possible input values (or 'x-values') for which the function is defined. In the video, the domain of the sine and cosine functions is discussed as all real numbers, since any angle can be plugged into these functions, whether positive, negative, or fractional multiples of pi.

💡Range

The range of a function is the set of all possible output values (or 'y-values') that result from the input values in the domain. The video explains that for sine and cosine, the range is from -1 to 1, inclusive, which reflects the vertical limits of the unit circle's coordinates.

💡Periodic

A periodic function is one that repeats its values at regular intervals or periods. The video emphasizes that trigonometric functions are periodic with a period of 2π radians, meaning that after every 2π radians, the function values repeat. This property is key to graphing and understanding the behavior of these functions over time.

💡Transformations

Transformations refer to the various ways a function can be altered, such as stretching, shrinking, shifting, or reflecting. In the video, transformations are discussed in the context of modifying the amplitude, period, and phase of trig functions. These transformations are crucial for adjusting the shape and position of the graphs of trigonometric functions.

💡Amplitude

The amplitude of a trigonometric function is the distance from the maximum value of the function to its midline. The video explains how multiplying the function by a coefficient (A) affects the amplitude, stretching or shrinking the function vertically. For example, y = 2sin(x) has an amplitude of 2, meaning it oscillates between -2 and 2.

💡Phase shift

A phase shift is a horizontal shift in the graph of a function. The video discusses how adding or subtracting a value inside the function (like BX + C) affects the phase shift. For instance, y = sin(x - π/3) represents a phase shift to the right by π/3 radians because the sine function starts at a different point on the x-axis.

💡Vertical shift

A vertical shift moves the graph of a function up or down without affecting its shape. In the video, adding a constant to the function (like y = sin(x) + C) results in a vertical shift. If C is positive, the graph moves up, and if C is negative, it moves down.

💡Asymptotes

Asymptotes are lines that a function approaches but never touches. In the context of the video, tangent functions have vertical asymptotes where the function is undefined, such as when cosine is zero. The video explains that these asymptotes occur at x = π/2 and x = 3π/2, where the tangent function approaches infinity.

💡Cosecant, Secant, Cotangent

These are the reciprocal functions of sine, cosine, and tangent, respectively. The video briefly mentions these functions, noting that they also have periodic behavior but with different graphical representations. For example, cosecant is the reciprocal of sine, and its graph has vertical asymptotes where sine is zero.

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.