To these may be added,

versin A = I cos A,

(9)

which follows from the fact that AP = AC - CP. By means of the equations established in this section the value of any trigonometrical magnitude may be expressed in terms of any other.

EXAMPLES.

1. Express sin A in terms of cos 4.

From equation (1) it follows that sin2 A = 1 - cosaA, and therefore,

2. Express tan A in terms of cos A.

Ans. sin A = V(I — cos2A).

sin A

cos A'

3. Express cos A in terms of tan A. From equation (6) and (2), we find

Ans. cos A =

I

V(1+tan 4)*

N. B.-This result might be obtained by solving the answer to

the last equation for cos A.

4. Express cosec A in terms of cos A.

3. Complemental and supplemental angles. DEFINITIONS:-The complement of an angle is the difference between it and a right angle.

The supplement of an angle is the difference between it and two right angles.

Thus the complement of 60° is 30°; of 49° 32', is, 40° 28'. The supplement of 60° is 120°; of 29° 37', is 150° 23'.

The following relations exist between the sines and cosines of complemental angles :

The sine of an angle is equal to the cosine of its complement.

The cosine of an angle is equal to the sine of its complement.

Let ACB be any angle, and ACE (fig. 3) its complement, then if BP and EQ be drawn perpendicular to AC, the triangles BCP and ECQ are equal in every respect, for since QEC is the complement of ACE, it is equal to ACB; the angles CPB and CQE are equal, both being right, and BC is equal to CE, both being radii; therefore (Euc. Book I. Prop. XXVI.) BP = CQ, i. e. the sine of an angle is equal to the cosine of its complement; and CP EQ, i. e. the cosine of an angle is equal to the sine of its complement.

=

The following relations exist between the sines and cosines of supplemental angles:

The sine of an angle is equal to the sine of its supplement.

The cosine of an angle is equal to the cosine of its supplement, but is of opposite sign.

Let ACB be any angle, and ACB' (fig. 3) its supplement, then as B'CA' is the supplement of ACB', it is equal to ACB; from which it follows (Euclid, Book Í. Prop. xxvi.) that the right-angled triangles BPC and B'P'C are equal in every respect, and therefore BP = BP', i. e. the sine of an angle is equal to the sine of its supplement; and CP = CP, i. e. the cosine of an angle is equal to the cosine of its supplement, and of opposite sign; because these lines lie, one on the right hand, the other on the left hand of the centre.

In the foregoing proof a principle is taken for granted which is assumed by mathematicians, to enable them to determine the position, as well as the magnitude of lines. It may be thus stated:Let two right lines (fig. 4), AA' and DD' intersect in C, then if C be considered as the origin from which all lines are measured, every line which is measured on AA' is positive or +, if it be situated to the right of C, negative or, if to the left. Every line measured on DD' is positive or +, if it lie above C, negative or —, if below C. Thus, if from M parallels be drawn to AA' and DD' respectively, CP is +, because it lies to the right of C; MP is + because it is measured by its equal CQ, which lies above C. Upon the same principle, CP and N'P are -, M'P' is +, and NP

The relation between the sines and cosines of complemental angles may be expressed by the equations:

sin A = cos (π – A),

cos Asin (-A).

(10)

(11)

If we divide the former equation by the latter, we obtain, by equations (4) and (5)

The tangent of an angle is equal to the cotangent of its complement.

The cotangent of an angle is equal to the tangent of its complement.

The relation between the sines and cosines of supplemental angles may be expressed by the equations:

4. Trigonometrical tables. It is to be remembered that the trigonometrical magnitudes defined in the first section as lines are, in reality, numbers; this follows from the supposition that the radius of the circle is equal to the linear unit. The following particular cases serve to place this in a clear point of view:

Let it be required to find the values of the sine, cosine, tangent, &c. of the angle 45°, which is equal to half a right angle; by equation (1),

I = sin3 45° + cos3 45°.

But as the triangle BCP is in this case (fig. 5) a right-angled isosceles triangle, it is evident that sin 45° cos 45°, and therefore,

=

and therefore,

I = 2 sin2 45°,
I = 2 cos2 45°;

sin2 45° = cos2 45° = •

If the square root of or 0.5 be extracted to five places of decimals, we find that

sin 45° = cos 45° = 0.70710.

As the sine and cosine are equal, it follows from equations (4) and (5) that

It also appears from (2) and (3), and the values just found for the tangent and cotangent, that

sec2 45° = cosec2 45° = 2;

and therefore, extracting √2 to five places of decimals, sec 45° = cosec 45° = 1.41421.

Let it be required to find the values of the sine, cosine, &c., of 60°. This angle is two-thirds of 90°, or of one right angle, from which it follows that the triangle ACB (fig. 6) is equilateral, and therefore that the radius AC is bisected in P; but as the radius = 1, CP; and therefore

cos 60° = = 0.50000.

From this value of the cosine we obtain that of the sine, as in Example 1. p. 9:

sin 60° = √(1 - 4) = √ };

=

extracting the square root of 0.75, to five places,

sin 60° 0.86602.

Let it be required to find the sine and cosine of 30°. As 30° is the complement of 60°, it follows from section 3, that

sin 30° = 0.50000,

cos 30° = 0.86602.

These cases are sufficient to show, that trigonometrical magnitudes are numbers, which are capable of being calculated from geometrical principles, and accordingly, tables, called Tables of Natural Sines, have been computed, in which the values of the sines, cosines, tangents, secants, &c., of every degree and minute in the quadrant are registered.

The statement of the method by which such tables are constructed is unsuited to the present treatise. The mode of using them in computation is explained in the Appendix.