SCHAUM’S Outline® Trigonometry With Calculator-Based Solutions 5th Edition

CHAPTER 13
Inverses of Trigonometric Functions

13.1 Inverse Trigonometric Relations

The equation

defines a unique value of x for each given angle y. But when x is given, the equation may have no solution or many solutions. For example: if , there is no solution, since the sine of an angle never exceeds 1. If there are many solutions , 150°, 390°, 510°, –210°, –330°, ….

In spite of the use of the word arc, (2) is to be interpreted as stating that “y is an angle whose sine is x.” Similarly we shall write y = arccos x if x = cos y, y = arctan x if x = tan y, etc.

The notation y = sin ^-1 x, y = cos ^-1 x, etc. (to be read “inverse sine of x, inverse cosine of x,” etc.) is also used but sin ^-1 x may be confused with 1/sin x = (sin x)^-1, so care in writing negative exponents for trigonometric functions is needed.

13.2 Graphs of the Inverse Trigonometric Relations

The graph of y = arcsin x = is the graph of x sin y and differs from the graph of y = sin x of Chap. 7 in that the roles of x and y are interchanged. Thus, the graph of y = arcsin x is a sine curve drawn on the y axis instead of the x axis.

Similarly the graphs of the remaining inverse trigonometric relations are those of the corresponding trigonometric functions, except that the roles of x and y are interchanged. (See Fig. 13.1)

Fig. 13.1

13.3 Inverse Trigonometric Functions

It is sometimes necessary to consider the inverse trigonometric relations as functions (i.e., one value of y corresponding to each admissible value of x). To do this, we agree to select one out of the many angles corresponding to the given value of x. For example, when we agree to select the value , and when we agree to select the value . This selected value is called the principal value of arcsin x. When only the principal value is called for, we write Arcsin x, Arccos x, etc. Alternative notation for the principal value of the inverses of the trigonometric functions is Sin^-1 x, Cos ^-1 x, Tan ^-1 x, etc. The portions of the graphs on which the principal values of each of the inverse trigonometric relations lie are shown in Fig. 13.1(a) to (f) by a heavier line.

When x is positive or zero and the inverse function exists, the principal value is defined as that value of y which lies between 0 and inclusive.

EXAMPLE 13.1 (a) Arcsin since and .

(b) Arccos since and .

When x is negative and the inverse function exists, the principal value is defined as follows:

EXAMPLE 13.2

13.4 Principal-Value Range

Authors vary in defining the principal values of the inverse functions when x is negative. The definitions given are the most convenient for calculus. In many calculus textbooks, the inverse of a trigonometric function is defined as the principal-valued inverse, and no capital letter is used in the notation. This generally causes no problem in a calculus class.

13.5 General Values of Inverse Trigonometric Relations

Let y be an inverse trigonometric relation of x. Since the value of a trigonometric relation of y is known, two positions are determined in general for the terminal side of the angle y (see Chap. 2). Let y₁ and y₂ be angles determined by the two positions of the terminal side. Then the totality of values of y consists of the angles y₁ and y₂, together with all angles coterminal with them, that is,

where n is any positive or negative integer or zero.

One of the values y₁ or y₂ may always be taken as the principal value of the inverse trigonometric function.

EXAMPLE 13.3 Write expressions for the general value of (a) arcsin 1/2, (b) arccos (–1), and (c) arctan (–1).

(a) The principal value of arcsin 1/2 is π/6, and a second value (not coterminal with the principal value) is 5π/6. The general value of arcsin 1/2 is given by

where n is any positive or negative integer or zero.

(b) The principal value is π and there is no other value not coterminal with it. Thus, the general value is given by , where n is a positive or negative integer or zero.

(c) The principal value is –π/4, and a second value (not coterminal with the principal value) is 3π/4. Thus, the general value is given by

where n is a positive or negative integer or zero.

SOLVED PROBLEMS

13.1 Find the principal value of each of the following.

(a) Arcsin

(b) Arccos

(d) Arccot

(e) Arccos

(f) Arccsc

(g) Arccos

(h) Arcsin

(i) Arctan

(j) Arccot

(k) Arccos

(l) Arccsc

13.2 Express the principal value of each of the following to the nearest minute or to the nearest hundredth of a degree.

(a) Arcsin or 19.47°

(b) Arccos or 66.42°

(d) Arccot or 40.10°

(e) Arccos or 14.39°

(f) Arccsc or 41.53°

(g) Arcsin or –40.08°

(h) Arccos or 116.87°

(i) Arctan or –55.22°

(j) Arccot or 126.28°

(k) Arccos or 145.97°

(l) Arccsc or –13.80°

13.3 Verify each of the following.

13.4 Verify each of the following.

13.5 Evaluate each of the following:

(a) cos (Arcsin 3/5),

(b) sin [Arccos (–2/3)],

(a) Let ; then , θ being a first-quadrant angle. From Fig. 13.2(a),

Fig. 13.2

(b) Let ; then , θ being a second-quadrant angle. From Fig. 13.2(b),

13.6 Evaluate .

Let

and

Then and , θ and φ being first-quadrant angles. From Fig. 13.3(a) and (b),

Fig. 13.3

13.7 Evaluate .

Let

and

Then and , θ and φ being first-quadrant angles. From Fig. 13.4(a) and (b),

Fig. 13.4

13.8 Evaluate sin (2 Arctan 3).

Let ; then , θ being a first-quadrant angle. From Fig. 13.5,

Fig. 13.5

13.9 Show that .

Let and then and each angle terminating in the first quadrant. We are to show that or, taking the sines of both members, that .

From Fig. 13.6(a) and (b),

Fig. 13.6

13.10 Show that .

Let θ = Arctan 1/2 and φ = Arctan 4/3; then and .

We are to show that or, taking the tangents of both members, that .

Now .

13.11 Show that .

Let θ = Arcsin 77/85, φ = Arcsin 3/5, and ψ = Arccos 15/17; then , , and , each angle terminating in the first-quadrant. Taking the sine of both members of the given relation, we are to show that . From Fig. 13.7(a), (b), and (c),

Fig. 13.7

13.12 Show that Arccot 43/32 - Arctan 1/4 = Arccos 12/13.

Let θ = Arccot 43/32, φ = Arctan 1/4, and ψ = Arccos 12/13; then , , and , each angle terminating in the first-quadrant. Taking the tangent of both members of the given relation, we are to show that . From Fig. 13.8, .

Fig. 13.8

13.13 Show that Arctan 1/2 + Arctan 1/5 + Arctan 1/8 = π/4.

We shall show that Arctan 1/2 + Arctan 1/5 = π/4 – Arctan 1/8.

and

13.14 Show that

Let θ = Arctan 1/3, φ = Arctan 1/7, , and then , , and each angle terminating in the first quadrant.

Taking the tangent of both members of the given relation, we are to show that

Now

and, using Fig. 13.9(a) and (b),

Fig. 13.9

13.15 Find the general value of each of the following.

where n is a positive or negative integer or zero.

13.16 Show that the general value of (a) Arcsin

(b)

where n is any positive or negative integer or zero.

(a) Let θ = Arcsin x. Then since , all values of Arcsin x are given by

Now, when (that is, n is an even integer), (1) may be written as ; and when , (that is, n is an odd integer), (2) may be written as . Thus, Arcsin Arcsin x, where n is any positive or negative integer or zero.

(b) Let θ = Arccos x. Then since , all values of Arccos x are given by and or Arccos x, where n is any positive or negative integer or zero.

(c) Let θ = Arctan x. Then since , all values of Arctan x are given by and or, as in (a), by nπ + Arctan x, where n is any positive or negative integer or zero.