§ 28. Relations between sine and cosine, tangent and secant, cotangent and cosecant.

By § 21 e, we have (Figs. 22-25) x2 + y2= r2, a formula which holds for all the figures since the squares of x, y, and r are positive in all cases; and from this we get:

The above formulas enable us to solve the following problems:

To find cos q when sin q is given, and the converse.

In order, however, to solve these problems completely, it is also necessary to know to what quadrant q belongs. When, for example, cos q is given, we have from [4] sin q= cos2q, which does not, on account of the double sign of the radical, completely determine sin q; but, when we also know to what quadrant & belongs, we know by § 25 the sign of its sine, and can therefore completely determine sin . It should be noted that formulas [1]-[6] enable us, when any one function is given, to find all the others. The student is advised, before reading farther, to do examples 1-9 at the end of this chapter.

§ 29. To represent the trigonometric functions by straight lines.

In order that the trigonometric functions, or ratios, may be represented by straight lines, the triangle of reference must be so taken for each ratio that the denominator of the ratio shall be unity.

In Figs. 26-29, let q, or the angle xOP, be an angle of

the 1st, 2d, 3d, and 4th quadrants respectively. From o as a centre, with a radius unity, describe a circle.

be unity, we will, in each figure, construct the triangle of

reference оCB, in which the hypothenuse oв is equal to

gent at A, we have the triangle of reference OAT, in which

A, draw a geometric tangent which will cut oв produced in some point as r'; from T′ let fall a perpendicular upon x'x. The triangle of reference OLT' thus formed will be a triangle in which the perpendicular LT' OA'= 1, and the base OL= A'T′.

=

oc, cb, at, ot, A'T', or', and CA are positive.

In Fig. 27,

CB, OT', and CA are positive,

and oc, at, or, and a'r' are negative.

In Fig. 28,

AT, A'T', and CA are positive, and CB, OC, OT, and or' are negative.

In Fig. 29,

oc, or, and ca are positive,

and CB, AT, A'T', and or' are negative.

Putting the above results in words, we have the following corollaries to our original definitions (v. § 24) : —

In a circle whose radius is unity, the trigonometric functions of an angle or arc may be represented by the following lines:

The sine, by the line which measures the perpendicular distance of the end of the arc from* the initial line.

The cosine, by the line drawn from the centre of the circle to the foot of the sine.

The tangent, by that part of the geometric tangent at the beginning of the arc which is drawn from the point of tangency to the terminal line.

The cotangent, by that part of the geometric tangent at a point+90° from the beginning of the arc which is drawn from the point of tangency to the terminal line.

The secant, by the line drawn from the centre of the circle to the end of the tangent.

The cosecant, by the line drawn from the centre of the circle to the end of the cotangent.

The versed sine, by the line drawn from the foot of the sine to the beginning of the arc.

What has been said in this section should not lead the student to regard the trigonometric functions as absolute lines. The trigonometric functions are not the lines themselves, but are merely the numbers which denote the ratios of the lines to the arbitrary unit which we choose as a radius.

*The word "from" indicates the direction, and therefore the sign of the line.

§ 30. In any circle the radius of which is unity, the sine of any arc is one-half the chord of twice the arc.

In Fig. 30 let the angle B'Oв or the arc B'B

= 2 p. Draw OA so as to bisect the angle or 2q; draw the chord B'B. Since OA bisects the angle B'Oв, we know from geometry that it is perpendicular to A and bisects the chord B'B. .*. CB = B'B = chord 2 g.

§ 31* Relations between the functions of 9 ± k 360°,

*A treatment of the subject of §§ 31, 32, and 33, covering explicitly all possible values of , is given in appendix I., which the teacher may at his discretion substitute for these sections.