Section 5.2: Exponential and Trigonometric Functions

In the previous section, we introduced the arithmetic of complex numbers.  This means that we can plug complex numbers into any formula that is defined in terms of arithmetic operations.  In particular, while we have not used the formula in computations, the natural exponential function, exp(t) = e^t, is given by such a formula and it makes sense to talk about the function C(t) = exp(t i) = e^{t i}.  All we can say right now about the function C is that its domain is the set of real numbers and its range is contained in the set of complex numbers.  In this section, we will discuss the function C in great detail.  We will then focus specifically on the related composite functions  and ,  We will see that they all have very nice geometric interpretations, related to the geometry of circles and triangles.

Complex Numbers, Exponentials, and the Unit Circle

A number of interesting properties of the function C follow directly from the rules of exponents and the geometry of the complex plane.  For example, since the conjugate of C(t) = e^{t i}, is  , the product .  This means that the distance of C(t) from the origin, .  In other words, the range of C(t) is limited to the circle of radius 1 centered at the origin, the so-called "unit" circle:

This is why we name this function C, for the "circle" function.  Although most values of C are difficult to compute, we can easily see that C(0) = e^{0 i} = e⁰ = 1. 

The real and complex parts of C(t) = e^{t i} give two real-valued functions, historically called cos(t) and sin(t), respectively, which give the horizontal and vertical coordinates of C(t).  That is, they are defined by Euler's formula, C(t) = e^{t i} = cos(t) + sin(t) i and may be pictured geometrically, as follows:

Note: Since we can show that C is given by the formula , algebraically, they are given by the formulas and , respectively.  

Since XFunctions has sin and cos as built-in functions, we can use them to plot C(t) as t goes from 0 to 10, as in the Example in the following applet:

As we mentioned earlier, C(t) "starts" at t = 0 with C(0) = 1.  If you hit the "Trace" button, you can see that as t increases, the value of C(t) goes counter-clockwise around the unit circle.  If you experiment by changing the value of "tmax" in XFunctions in the plot of C, you can estimate the following table of values for C(t): 

Notice that -1 = C(t) = e^{t i} = cos(t) + sin(t) i implies that cos(t) = -1 and sin(t) = 0.  Using the "x =" box in XFunctions, or your calculator, by trial and error you can estimate this value of t more precisely.  You should recognize the more exact value of t = 3.1415... as p!  

As you could see when XFunctions plotted C, the point C(t) moves at a constant speed around the unit circle, so the values of t in our table should be all equally spaced.  This suggests that a more accurate table would be:

Because C(t) moves on a circle of radius 1, with circumference equal to 2p, you can see that:

The value of a positive input, t > 0, is precisely equal to the distance traveled around the circle from C(0) = 1. 

Notice also how how this distance traveled clockwise, t, corresponds exactly to the angle traversed from the positive horizontal axis.  Thus, it is convenient to define a new unit of measurement for angles, called radian measure with relates directly to the value of t.  We can define this angle measure by comparison with two other common measures, as in the following table:

Although degree measure has been used for thousands of years, since ancient Babylonian times, its main advantage is the fact that the degree measure of certain commonly used angles are all integers.  While there are various theories behind the choice of 360 as the basis for degree measure, no one really knows.  In any case, since both "rotations" and "radians" are defined geometrically, they are both superior choices of units.  Just as e is the "natural" base for logarithms, because of the relationship between C(t) and the unit circle, radian measure is a more "natural" unit of measure for angles.  It is superior even to "rotations", since it directly relates angle and length measurements.  

It is convenient to think of t as an angle measurement (in radians).  This allows us to give a direct geometric description for C(t):

C(t) = e^{t i} = cos(t) + sin(t) i is the point on the unit circle in the complex plane which is rotated t radians around the origin / located t units along the circle from 1, as in the following picture:

We have seen that, for positive values of t, we move counter-clockwise around the circle.  As you might expect, for negative values of t, we must move clockwise.  You can see this by tracing the Example in the following applet:

Note: This shows C(-t) starting at C(0) = 1; as t increases positively, we can see the value of C for ever "larger" negative inputs.

Notice that, since this function repeats itself each time we wind once more around the circle, we can extend the table indefinitely in both directions:

We say that such a repeating function is periodic.  Since it repeats each 2p  units, we say that it has period 2p.  

Practice working with radian measure and the complex exponential function, C(t) = e^{t i} = cos(t) + sin(t) i, by completing the following Exercises.  

Introduction to Trigonometric Functions

Now that we have a good understanding of the "circle" function, C(t), we want to move on to discuss the functions cos and sin, and the other related trigonometric functions.  Our geometric description for C(t):

leads to geometric descriptions for cos and sin:

cos(t) and sin(t) are the horizontal and vertical coordinates of the point on the unit circle in the complex plane which is rotated t radians around the origin / located t units along the circle from 1.

We can deduce the shape of the graphs of cos and sin, respectively, by focusing on the horizontal and vertical coordinates of C(t):

Looking at the plot of C(t), given as before in the previous applet, verify that the following descriptions are true:

• Since cos(t) is the horizontal/real coordinate of C(t):

  • cos starts at 1 (i.e., cos(0) = 1) and

  • gets smaller until it reaches a minimum of -1 (passing through 0 along the way), then

  • gets larger until it regains its maximum of 1 (passing through 0 along the way), and

  • repeats the same values over and over again.

• Since sin(t) is the vertical/imaginary coordinate of C(t):

  • sin starts at 0 (i.e., sin(0) = 0) and

  • gets larger until it reaches a maximum of 1, then

  • gets smaller until it reaches a minimum of -1 (passing through 0 along the way), then

  • goes back to 0 and repeats the same values over and over again.

You can quickly verify these observations by looking directly at the plots of cos and sin in XFunctions:

From our table of values for C(t), we can immediately write down tables of values for cos and sin:

which, like C, must be periodic and repeat indefinitely for positive and negative values of t.

From these two functions, we can define four other functions: csc, sec, cot, tan.  They are given by the following formulas: 

csc(t) = 1/sin(t)
sec(t) = 1/cos(t)
cot(t) = cos(t)/sin(t)
tan(t) = sin(t)/cos(t)

with the corresponding tables of values:

You can compare these tables with the corresponding plots given in XFunctions:

Note: While we present these functions all together, they were actually developed hundreds years apart to solve rather different applied problems; sin was developed by Ptolemy around 100 AD to make measurements in circles, the Arabs began to use around 800 AD tan while studying shadows (e.g., for constructing sundials and other astronomical calculations), and sec was not extensively calculated until it was used by 15th century Europeans as an aid to navigation.  With the wisdom of hindsight, we can now easily see how these functions are directly related.

The names for these functions have an interesting history.  They are all shortened forms of longer, more meaningful names:

Notice how the functions naturally come in pairs, as a function (on the right) and its "co-" function (on the left).  The suffix, "co-" stands for "complementary", as in complementary angles.  You may remember from Geometry that complementary angles are angles which add to a right angle (i.e., p/2 radians = 90º).  For example, "cosine" stands for "sine of the complementary angle".  Since the complement of an angle t (in radians) would be the angle p/2 - t, the names suggest that:

cos(t) = sin(p/2 - t)
csc(t) = sec(p/2 - t)
cot(t) = tan(p/2 - t)

For example, from our graphing principles, this means that the graph of cos(t) = sin(p/2 - t) can be obtained by taking the graph of sin, shifting left by p/2 and reflecting horizontally:

This gives the following table of values:

which, when we extend periodically:

we recognize as equal to that of the cos function.

We may also verify this directly by visually comparing the plot of sin(p/2 - t) with cos(t), as in the following Example in XFunctions:

The names secant and tangent come via a little geometry with the circle picture.  In general, a tangent (from the Latin tangere = "to touch") line is a line that barely touches a figure at one point, while a secant (from the Latin secare = "to cut") line cuts across the figure.  In the following picture:

OT is a secant line, while AT is a tangent line.  By similar triangles, we obtain a geometric interpretation for these two functions, when 0 < t < p/2:

OT = OT/1 = 1/cos(t) = sec(t) and AT = AT/1 = sin(t)/cos(t) = tan(t).

In other words, when 0 < t < p/2, the secant function gives the length of the secant line, and the tangent function gives the length of the tangent line.

Finally, the story behind the name "sine" is the most interesting.  It comes from a series of mistranslations of the Indian word jya for "chord", which refers to any line segment whose endpoints lie on a circle.  The first partial tables for the sine function were computed in around 500 AD by Aryabhata as half the length of the chord CS in the following picture:

as the angle t varied. The Indian phrase jya-ardha = "chord-half" for this function was abbreviated simply to jya.  However, around 700 AD this term was transliterated into the meaningless Arabic word jiba.  Unfortunately, this led European scholars, who were trying to translate the works of Arabic mathematicians, such as Abu'l-Wafa, to mistake jiba for the word jaib, meaning "fold".  Around 1000 AD, European mathematicians such as, Fibonacci, then translated jaib into the equivalent Latin word sinus, meaning "fold", "hollow", or "cavity", from which we get the name "sine"!

In closing, it is interesting to observe that, using the definitions and the co-function identities, we can write every trigonometric functions in terms of the sin function alone:

cos(t) = sin(p/2 - t)
csc(t) = 1/sin(t)
sec(t) = 1/sin(p/2 - t)
cot(t) = sin(p/2 - t)/sin(t)
tan(t) = sin(t)/sin(p/2 - t)

In other words, from a functional perspective, trigonometry is "simply" the study of the sin function!  In the next section, we will see that this function has a great number of surprising algebraic properties.

Practice estimating values for the various trigonometric functions by completing the following Exercises.  

Go to The Algebra of Trigonometric Functions

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Send questions or comments to jwicks@northpark.edu

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