47. The SINE of an angle is the ratio of the opposite side to the hypothenuse.

Thus, in any right-angled triangle, A B C,

if the sides be denoted by p, b, h, we shall

have,

B

48. The TANGENT of an angle is the ratio of the opposite side to the adjacent side.

49. The SECANT of an angle is the ratio of the hypothenuse to the adjacent side.

50. The COSINE, COTANGENT, and COSECANT of an angle are respectively the SINE, TANGENT, and SECANT of its complement. Hence, since the acute angles of a right-angled triangle are complements one of the other (Art. 44), we have, according to the definitions,

that the cosecant, cotangent, and cosine of an angle are respectively the reciprocals of the sine, tangent, and secant of the angle.

That is,

b

of?

h

we see

h' p

sin A cosec A= 1, cos A sec A1, tan A cot A= 1.

52. If the cosine of A be subtracted from unity, the remainder is called the versed sine of A; if the sine of A be subtracted from unity, the remainder is called the coversed sine of A; and if the cosine of A be added to unity, the sum is called the suversed sine of A. Hence,

vers 4=1—cos A, covers 4 = 1 — sin ▲,

suvers A=1+cos A. j

(6)

53. The values of trigonometric ratios remain the same so long as the angle continues the same.

Let BA C be any angle; in A B take any point, B, and draw BC perpendicular to A C; also take any other point, B', and draw B'C' perpendicular to A C. Then, since

the triangles ABC, AB' C' are similar, their
sides have to one another the same ratio
(Geom., Art. 210), and therefore sin A, tan A, A
&c. will have the same values, whether ABC

B'B

C' C

or AB'C' be the triangle by the sides of which they are expressed. It is also evident that their values would change with a change of the angle. Hence,

The trigonometric ratios determine the angles, and conversely; that is, any determinate values being given for the one, determinate values can be found for the other.

54. The terms sine, tangent, secant, &c., were formerly* considered to be functions of an arc, and denoted certain trigonometric lines.

A Cot.

D'S

sec

Thus, let O be the centre of any circle, AA" its diameter, and AB any arc; draw the radius O A' at right angles to AA", and draw tangents to the circle at the points A and A'; produce OB to meet the first tangent in T and the second tangent in T"; draw BD perpendicular to O A, and B D' perpendicular to OA'. Then, by the old definitions, the lines of the figure are considered to

Suvers

Sec

Cos. Vers

D

A

* "The modern method has now completely superseded the ancient method

in English works."- Todhunter's Trigonometry, p. 49.

be the functions of the arc A B. BD is the sine of the arc AB, OD its cosine, A T its tangent, A'T' its cotangent, O T its secant, O T'' its cosecant, AD its versed sine, A' D' its coversed sine, and A" D its suversed sine. is the chord of the arc A B.

Also the line joining A and B

That is, in the circle whose radius is unity;

1

The SINE of an arc, or of the angle measured by that arc, is the perpendicular let fall from one extremity of the arc, upon the diameter passing through the other extremity.

The COSINE is the distance from the centre to the foot of the sine.

The trigonometric TANGENT is that part of the tangent touching one extremity of the arc, which is intercepted between that extremity and the radius produced passing through the other extremity.

The SECANT is that part of the radius produced which is intercepted between the centre and the tangent.

The VERSED SINE is that part of the diameter intercepted between the foot of the sine and the origin of the arc.

The COTANGENT, COSECANT, and COVERSED SINE are tangent, secant, and versed sine, respectively, of the complement of an arc or angle. The cosine is also equal to the sine of the complement, as ODD' B.

The SUVERSED SINE is that part of the diameter which remains after taking away the versed sine, or it is the versed sine of the supplement.

55. If the radius of the circle be unity, the numerical value of the sine and other trigonometric functions is the same in both the old and new systems, for

sin A OB

Ᏼ Ꭰ
OB'

sin AB BD.

=

But O B is the radius of the circle, and denoting it by r, we have

In like manner it may be shown, that similar results hold for

all the other trigonometric functions. Hence any formula expressed in the old system may be immediately converted into a formula expressed in the new system, by supposing the radius of the circle to be equal to unity.

56. The sine, cosine, tangent, and cotangent constitute the primary class of trigonometric ratios, as they are by far most frequently used; and the others form a subordinate class, the employment of which is occasionally attended with convenience. They are collected, for more ready reference, in the following

57. To find the cOSINE of an angle by means of its sine.

From the right-angled triangle ABC

(Geom., Prop. XI. Bk. IV.) we have

p2 + b2 = h2.

Dividing both sides of the equation by h2 A

B

h

P

in which "sin3 A" denotes "the square of the sine of A.” 58. To find the SINE of an angle by means of its cosine. Since, by (8), sin2 A+ cos2 A = 1,

59. To find the TANGENT and COTANGENT of an angle by means of the sine and cosine.

By (2) we have

60. To find the SECANT and COSECANT of an angle by means

of the tangent.