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David Terr
Ph.D. Math, UC Berkeley

 

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4.7. Inverse Trigonometic Functions

Video Lecture on Arcsine

 

Thus far in this chapter, we have looked at the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) in great detail. Also useful are the inverses of these functions, namely arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant. We investigate each of these function in turn - tabulating, graphing, and pointing out some basic properties of each one.

The most basic inverse trigonometric function is the arcsine, abbreviated arcsin x. (Some texts use the notation sin-1x for this function and similar notation for the other inverse trigonometric functions, but we prefer the former.) Since sin x is not one-to-one, in defining its inverse we must first restrict its domain to a region in which it is one-to-one. Looking at Figure 4.5.2, we see that sin x is one-to-one, and in fact strictly increasing, over the inteval (-π/2,π/2). Thus we restrict the domain to this region, which becomes the range of arcsin x. Now to define arcsin x, we simply do what we do in defining the inverse of any function - we swap the roles of x and y. Thus, for all x in the interval [-1,1] (the range of sin x and the domain of arcsin x), the value of arcsin x is defined to be the number y such that sin y = x. Thus, to tabulate arcsin x, we simply swap the first two columns in Table 4.5.1. We must be careful though. We can only use the rows of the table corresponding to values of x from 0 to π/2. To get the values of arcsin x for x from -π/2 to 0, we simply note that sin x is an odd function, whence arcsin x must also be odd. Thus we have

  • (4.7.1) arcsin(-x) = -arcsin x

for all x in the domain of the arcsine function. This formula allows us to easily compute the remaining table entries. With all this said, here is the resulting table:

x arcsin x
-1.000
-π/2 ≈ 1.571
-√3/2 ≈ -0.866
-π/3 ≈ -1.047
-√2/2 ≈ -0.707
-π/4 ≈ -0.785
-1/2 ≈ -0.500
-π/6 ≈ -0.524
0.000
0.000
1/2 = 0.500
π/6 ≈ 0.524
2/2 ≈ 0.707
π/4 ≈ 0.785
3/2 ≈ 0.866
π/3 ≈ 1.047
1.000
π/2 ≈ 1.571

arcsin Table 4.7.2

Now we plot a graph of these values, obtaining the following graph of the arcsine function.

graph of arcsin x

Figure 4.7.3: Graph of arcsin x

The arccosine function is defined similarly to the arcsine function. Once again, we must first choose a domain with which to restrict cos x to make it one-to-one. One easily sees from its graph that the interval [0,π] does the job. Below we tabulate and graph arccos x.

x arccos x
-1.000
π ≈ 3.142
-√3/2 ≈ -0.866
5π/6 ≈ 2.618
-√2/2 ≈ -0.707
3π/4 ≈ 2.356
-1/2 ≈ -0.500
2π/3 ≈ 2.094
0.000
π/2 ≈ 1.571
1/2 = 0.500
π/3 ≈ 1.047
2/2 ≈ 0.707
π/4 ≈ 0.785
3/2 ≈ 0.866
π/6 ≈ 0.524
1.000
0.000

arccos Table 4.7.4

As can be seen below, the graph of the arccosine function looks very similar to that of the arcsine function. These two graphs have the same shape, that of arccos x merely being reflected and shifted from that of arcsin x. In fact, the graphs of both of these functions are sinusoids, merely reflected about the diagonal line y=x from the graphs of sin x and cos x respectively. Note also that arccos x is strictly positive throughout its domain, its range being the interval [0,π].

graph of arccos x

Figure 4.7.5: Graph of arccos x

It should be noted that Equations (4.4.1) and (4.4.2) imply the following simple relationships between the arcsine and arccosine functions. We leave the proofs of these formuals as an exercise. (Actually, one is trivial to derive from the other.)

  • (4.7.6a) arccos x = π/2 - arcsin x
  • (4.7.6b) arcsin x = π/2 - arccos x

Like arcsine and arccosine, the arctangent and arccotangent functions have similar looking graphs. First we define arctan x by restricting the domain of tan x to the interval [-π/2, π/2], over which it is clearly one-to-one. This interval covers one of infinitely many similar branches of the function. Thus, the graph of the arctangent function consists of one branch of the tangent function turned sideways and reflected about the y-axis, or equivalently, reflected about the diagonal line y=x. In tabulating tan x, we make use of the fact that this function is odd, since its inverse is odd. Thus we have

  • (4.7.7) arctan(-x) = -arctan x
x arctan x
-√3≈-1.732
-π/3 ≈ -1.047
-1.000
-π/4 ≈ -0.785
-√3/3 = -0.577
-π/6 ≈ -0.524
0.000
0.000
3/3 = 0.577
π/6 ≈ 0.524
1.000
π/4 ≈ 0.785
3≈1.732
π/3 ≈ 1.047

arctan Table 4.7.8

Note that the graph of arctan x has two asymptotes, one at y=-π/2 and the other at y=π/2. These are indicated with dashed lines in the figure below.

graph of arctan x

Figure 4.7.9: Graph of arctan x

 

To graph the arccotangent function, we restrict the domain of cot x to the interval [0,π], where we see that it is one-to-one and consists of a single branch. Sparing the details of our calculations, we present the table and the graph of this function below.

x arccot x
-√3≈-1.732
5π/6 ≈ 2.618
-1.000
3π/4 ≈ 2.356
-√3/3 = -0.577
2π/3 ≈ 2.094
0.000
π/2 ≈ 1.571
3/3 = 0.577
π/3 ≈ 1.047
1.000
π/4 ≈ 0.785
3≈1.732
π/6 ≈ 0.524

arccot Table 4.7.10

graph of arccot x

Figure 4.7.11: Graph of arccot x

 

The remaining two inverse trigonometric functions, arcsecant and arccosecant, are the most obscure, but they are still occasionally used, so we present them as well for completeness. We select the domain [0,π] of sec x to define its inverse, arcsec x, and the domain [-π/2, π/2] of csc x to define arccsc x. Without further ado, we tabulate and graph these functions below.

x arcsec x arccsc x
-2.000
2π/3 ≈ 2.094
-π/6 ≈ -0.524
-√2≈ -1.414
3π/4 ≈ 2.356
-π/4 ≈ -0.785
-2√3/3 ≈ -1.155
5π/6 ≈ 2.618
-π/3 ≈ -1.047
-1.000
π ≈ 3.142
-π/2 ≈ -1.571
1.000
0.000
π/2 ≈ 1.571
2√3/3 ≈ 1.155
π/6 ≈ 0.524
π/3 ≈ 1.047
2≈ 1.414
π/4 ≈ 0.785
π/4 ≈ 0.785
2.000
π/3 ≈ 1.047
π/6 ≈ 0.524

arccot Table 4.7.12

 

graph of arcsec x

Figure 4.7.13: Graph of arcsec x

 

graph of arccsc x

Figure 4.7.14: Graph of arccsc x

It should be noted that the dashed lines in the above two graphs are not all asymptotes.Instead, the top dashed line in Figure 4.7.13 and both dashed lines in 4.7.14 indicate limits of the ranges of these functions.

 

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