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David Terr
Ph.D. Math, UC Berkeley

 

Home >> Pre-Calculus >> 4. Trigonometric Functions

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>> 4.7. Inverse Trigonometic Functions

 

4.6. Graphs of Other Trigonometic Functions

In the last section we looked at graphs of sine and cosine, the two most basic trigonometric functions. In this section we look at graphs of the other four trigonometric functions, namely tangent, cotangent, secant and cosecant. We start with the tangent function. By virtue of the equation tan x = sin x / cos x and cot x = 1 / tan x, it is easy to compute the values of these functions once we know sin x and cos x. Thus, it is straightforward to derive the following table. As usual, dashed entries mean the value of the corresponding functions is undefined at this value of the argument. Note that we only list values of x from 0 to π, since the periods of the tangent and cotangent functions are both equal to π, unlike sine and cosine, whose periods are 2π.

x tan x cot x
0.000
0.000
-
π/6 ≈ 0.524
3/3 = 0.577
3 ≈ 1.732
π/4 ≈ 0.785
1.000
1.000
π/3 ≈ 1.047
3≈1.732
3/3 = 0.577
π/2 ≈ 1.571
-
0.000
2π/3 ≈ 2.094
-√3≈-1.732
-√3/3 = -0.577
3π/4 ≈ 2.356
-1.000
-1.000
5π/6 ≈ 2.618
-√3/3 = -0.577
-√3≈-1.732
π ≈ 3.142
0.000
-

Table 4.6.1

 

Below are the graphs of tan x and cot x. Note that each of these graphs show singularities in the argument at intervals of π. Asymptotes of these functions at singular values are indicated with dashed vertical lines. Also note that these graphs have the same shape except that the graph of cot x is a reflection of the graph of tan x about the line x = π/4. Finally, we note that both tan x and cot x are odd functions, a fact which is easy to prove from our knowledge of the trigonometric functions.

graph of tan x

Figure 4.6.2: Graph of tan x

 

graph of cot x

Figure 4.6.3: Graph of cot x

 

It is also easy to tabulate values of the secant and cosecant functions by virtue of the formulas sec x = 1 / cos x and csc x = 1 / sin x. A table of these functions for x going from 0 to 2π is given below.

x sec x csc x
0.000
1.000
-
π/6 ≈ 0.524
2√3/3 ≈ 1.155
2.000
π/4 ≈ 0.785
2≈ 1.414
2 ≈ 1.414
π/3 ≈ 1.047
2.000
2√3/3 ≈ 1.155
π/2 ≈ 1.571
-
1.000
2π/3 ≈ 2.094
-2.000
2√3/3 ≈ 1.155
3π/4 ≈ 2.356
-√2≈ -1.414
2 ≈ 1.414
5π/6 ≈ 2.618
-2√3/3 ≈ -1.155
2.000
π ≈ 3.142
-1.000
-
7π/6 ≈3.665
-2√3/3 ≈ -1.155
-2.000
5π/4 ≈ 3.927
-√2≈ -1.414
-√2 ≈ -1.414
4π/3 ≈ 4.189
-2.000
-2√3/3 ≈ -1.155
3π/2 ≈ 4.712
-
-1.000
5π/3 ≈ 5.236
2.000
-2√3/3 ≈ -1.155
7π/4 ≈5.498
2≈1.414
-√2 ≈ -1.414
11π/6 ≈5.760
2√3/3 ≈ 1.155
-2.000
2π ≈ 6.283
0.000
-

Table 4.6.4

 

Below are the graphs of the secant and cosecant functions. Note that like tangent and cotangent, these functions have singularities, indicated by asymptotes. Also note that the range of each of these functions excludes the interval (-1,1). We also note that like sine and cosine, these functions are periodic with period 2π. Finally, we note that sec x is even and csc x is odd.

graph of sec x

Figure 4.6.5: Graph of sec x

 

graph of csc x

Figure 4.6.6: Graph of csc x

 

Home >> Pre-Calculus >> 4. Trigonometric Functions

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