-

86. Principal Values. We have seen (§ 79, p. 94), and it is also immediately evident from Fig. 76, that there is one and only one angle a between π/2 and +π/2* whose sine is x, if x lies between 1 and +1. We shall distinguish this one value from all other angles whose sine is x by capitalizing the initial letter A in the symbol arcsin; thus arcsin x has many values, but Arcsin x has one and only one value α, if x is any given number between - 1 and +1. We shall call this special value (= Arcsin x) the principal value of arcsin x.

Thus, if x=1/2, Arcsin x =π/6; if x - 1/2, Arcsin x = — π/6; etc. In Fig. 76, if x is represented by OK, a vertical line through K meets the curve in one and only one point P between y = -π/2 and y=+π/2.

Another value of arcsin x is (OB, Fig. 76) π Arcsin x, which is obtained by subtracting the principal value from π, since sin ( α) = sin α = x. = It follows that all arcsin x are congruent either to Arcsin x or to πthat is, all values of arcsin x are contained in one or the other of the forms

(1)

values of

Arcsin x;

Arcsin x2 nπ, or (π — Arcsin x) ± 2 nπ,

where n is a positive integer or zero.†

is π

=

For example, if x = 1/2, Arcsin x = π/6; but another value of arcsin x - π/6 5 π/6 (OB, Fig. 76). All other values are given by the forms π/6+2 nя or 5 π/6 ± 2 n. Thus, π/6+2π = 13 π/6 (OC, Fig. 76) and 5/6-2 π = — 7 π/6 are other values of arcsin x.

Similarly, arccos x has one and only one value ɑ between ‡ O and π (see II, § 79, and OA, Fig. 78) if x lies between 1 and +1. This value will be denoted by Arccos x and will be called the principal value of arccos x.

Another value of arccos x is

Arccos x (OB, Fig. 78) since cos (a) = cos α = x. All values of arccos x are contained in one or the other of the forms:

* That is, π/2 radians, or a right angle. Here and throughout this Chapter, radian measure is to be understood.

† Or, in one formula, aresin x = Aresin [(-1)"x] + nπ, n = 0, 1, 2, etc. Notice that values of y between π/2 and π/2 would not be enough to

include all values of cos y.

For example, Arccos (1/2) =T/3; another value of arccos (1/2) is -π/3; all others are given by π/32 nπ or r/3 ± 2 nπ. Thus, π/3 + 2 π = 5 π/3 (see OC, Fig. 78) is another value of arccos (1/2).

=

Finally, arctan x has one and only one value a (: Arctan æ) between -π π/2 and +π/2; and it will be called the principal value of arctan x. Another value of arctan x is + Arctan x, since tan (+ α) = tan α = x. All values of arctan x are contained in one or the other of the forms: Arctan x±2 nя, oг (+Arctan ) 2 nm, which are together equivalent to

Thus, Arctan √3 =π

=7π/6 and π

π/6; but other values of arctan √3 are π + π/6 π/6 π = − 5 π/6.

The principal values of the other inverse trigonometric functions are denoted by Arcctn x, Arcsec x, Arccsc x, etc., and are given in the following table:

* Or, in one formula, arccos x = Arccos [(-1)]+nï, n = 0, ±1, ±2, etc.

EXERCISES XXXIV.-INVERSE FUNCTIONS. GRAPHS

1. Plot the curve y = cos x from x = 0 to xπ by a construction

2. Plot the curve y=

tan x for the range -π π/2 < x <π/2, making use of the construction shown in the annexed figure: CA is one scale unit; ACB is the angle x radians; the Dordinate of point B is tan x.

3. On the graph of each of the following curves, mark in heavier ink the portion which corresponds to the principal value (see table above). = arcctn x ; (b) y = arcsec x; (c) y = arccsc x.

(a) y

4. Prove each of the following relations:

(a)

Arcsin (3/4) = — Arcsin (3/4).

let ß = Arcsin 3/4, then

[This can be seen from the graph of Arcsin x; proved as follows: Let a Arcsin (-3/4), then sin α = - – 3/4; sin ẞ 3/4, where both and ẞ lie between

sin (— ẞ) — sin ß = − 3/4. Hence sin α = sin

i.e. Arcsin (3/4) = - Arcsin 3/4.]

=

π/2 and +π/2. Now (— ẞ); whence α=

(b) Arcsin ( x) = — Arcsin x for - 1 < x < 1. (c) Arccos (- 2/5) = π — Arccos (2/5).

(d) Arccos (— x) = π — Arccos x for − 1≤ x ≤ 1. (e) Arcsin 2/3 + Arccos 2/3 = π/2.

(ƒ) Arcsin x + Arccos x = π/2 for −1≤x≤1.

(g) Arctan (1/2) + Arcetn (1/2) =π π/2.

(h) Arctan x + Arcctn x = π/2 for all values of x.

(i) Arcsin x = ± Arccos √1 − x2, according as x > 0 or x <0.

5. Show that Arctan x + Arctan y

and y are both between

1 and +1.

=

x

Ba

Arctan (+), provided x

6. Arctan x Arcsin x/√1 + x2 for all values of x.

7. Aretan 1/2 + Arctan 1/3

8. Aretan 1/4 + Arctan 1/13

9. Arctan 1/2 + Arctan 1/5 + Arctan 1/8 = π/4.

10. Arctan æ + Areetn (x + 1)= Arctan(x + x+1) for all x. 11. Arccos 1/√2 – Arcsin 1/√5 = Arctan 1/3.

12. Find the numerical values of the following: (a) cos (Arcsin 8/17).

(c) tan (Aretan 4/3 – Aretan 1/7).

(b) tan (Aresin 5/13).

87. Transcendental Equations. In §§ 77-83 equations occur which involve only trigonometric functions of the unknown. In equations which involve also algebraic or other functions, the unknown angle is generally measured in radians. The graphical method is usually the best. All such equations. are called Transcendental Equations.

Example 1.

Solve the equation x cos x. To solve this equation we must find a number x, such that the cosine of x radians is equal to the number x. Draw the graphs of the equations y = x and y = cos x; the solution of the equation is the value of x at the point of intersection of the two graphs. There is clearly only one such point, and at that point x= .74, approximately; a still more accurate value is x = .73908.

[HINT. Use logarithms to plot the curves y = 2*, etc.]

=

cos.x. logx sin x.

8. 10x
9. (2.7)

cos x.

=

sin x + cos x.

CHAPTER VIII

SPHERICAL TRIGONOMETRY

PART I. SUMMARY OF GEOMETRIC THEOREMS

88. Purpose. Spherical Trigonometry has for its primary purpose the determination of certain parts of a spherical triangle when other parts are known. This is accomplished by the use of the formulas of plane trigonometry and certain elementary propositions of solid geometry. Several useful propositions are collected here without proof for convenience in reference.

89. Some Useful Propositions of Solid Geometry. (1) A sphere is the locus of points equidistant from a fixed point O called its center. The constant distance is called the radius, and will be denoted in all that follows by R.

(2) If a sphere is cut by any plane, the intersection is a circle. If the plane passes through the center O, the radius r of the circle is equal to R, and the circle is called a great circle. Otherwise we have r < R, and the circle is called a small circle.

(3) A line perpendicular to a plane is perpendicular to every line of the plane through the foot of the perpendicular.

[We shall use also several other theorems regarding lines and planes that are practically self-evident.]

(4) A line through the center O perpendicular to the plane of a given circle cuts the sphere in two points P and P', which are called the poles of that circle. The line PP' is called the axis of that circle.

For example, the north and south poles of the earth are the poles of the equator.

(5) Through any two points A and B of a sphere there is at least one great circle, determined by the plane AOB; there is only one such great circle through A and B unless AOB is a straight line.