A

A

generally from sin A, must have four values, since each of them may correspond to any one of the four positions OP1, OP 2, OP3, OP of the revolving line, which have their sines and cosines all different.

4

17. If we know however the limits between which A lies sufficiently to determine in which of certain quadrants the revolving line corresponding to 4 falls, there is no

A

2

A

longer any ambiguity in determining the value of sin and

A

COS This we shall proceed to shew.

2

18. To trace the change in the sign of sin 4+

2

2

A

2

as varies from 0° to 360°, that is,

2

as A varies from zero to 720o.

Observe that it is only necessary to take notice of the

2

for the sign of the greater one will clearly determine the sign of the whole expression.

Now from -45° to +45° the cosine is greater than the sine and is positive.

A

2

2

From 45° to 135° the sine is greater than the cosine, and is positive;

A

.. sin

A

+ cos is positive,

2

A

COS is positive;

.. sin + Cos

2

2

=

√(I-sin A).

From 135° to 225° the cosine is greater numerically than the sine, and is negative;

A

COS = √(1 − sin A).

2

From 225° to 315° the sine is numerically greater than the cosine, and is negative;

.. sin

√(1 + sin A),

√(I-sin A).

{√(1 + sin A) — √(1 − sin A)},

2

2

E

H. T.

A

A

2

2

I

2

Example. Given sin 30°: find sin 150.

These might of course be found as follows:

sin 150=sin (45°-30°)=sin 45° cos 30°-cos 45° sin 30°, cos 15°= cos(45° — 30°) = cos 45° cos 30°+ sin 45° sin 30o, and the sines and cosines of 45° and 30° are known.

CHAPTER V.

DETERMINATION, A PRIORI, OF THE NUMBER OF VALUES WHICH ANY ASSIGNED TRIGONOMETRICAL FUNCTION MAY HAVE WHEN DETERMINED FROM ANY OTHER FUNCTION OF THE ANGLE, OR OF A MULTIPLE OR SUBMULTIPLE OF THE ANGLE.

TH

HESE articles are given to illustrate the employment of the general formulæ given in Chap. III. Arts. 7, 8, 9, for equisinal, equicosinal angles, &c.

1. To determine, à priori, how many values sin A will have when determined from cos A, which is supposed to be known.

If a be the circular measure of the least primary angle which has its cosine equal to cos A, any of the angles included in the formula 2nπ±a will have its cosine equal to cos A. Hence, in finding the value of sin A from cos A, we do not know which individual of this group we must take; for although the cosines of them all are the same, it does not follow that the sines are so. Hence all the values of sin A which can be got by taking all the angles whose cosine is cos A are included in the formula

sin (2nπα),

which is equal to sin (a) or sin a;

so that sin A, when determined from cos A, has two values equal in magnitude but of opposite signs. This corresponds to and explains Chap. II. Art. 7 (11).

2. Let tan A be given, to determine sin A.

Since all the angles which have a given tangent are included in the formula na+a, where a is the least positive

A

A

2

2

angle whose tangent is tan A, all the values which sin A can have when determined from tan A will be included in sin (n+a), which is equal to sin a, according as n is even or odd.

Hence sin A, when determined from tan A, has two values. Compare Chap. II. 7 (111).

3. Given cos A, to find how many values sin

A

A

and

2

COS

will have when expressed in terms of it.

2

Let a be the circular measure of the least primary angle whose cosine is cos A. Then all the angles, the cosines of which are equal to cos A, are included in the expression

when expressed in terms of cos A, will be included respec

for if n is even, they are equal to + sin

and to cos re

2

2

spectively;

α

2

and if n is odd, they are equal to sin and to-cos

respectively.

α

2

This corresponds to and explains the results (11) and (12) in Chap. IV.

4. Given sin A, to determine, à priori, how many

Let a be the circular measure of the least primary angle, the sine of which is equal to sin A. Then all the angles whose sines are equal to sin A are included in the general formula