denominators of the ratios shall be equal to this unit. The most convenient way to do this is as follows:

About a point 0 (Fig. 3) as a centre, with a radius equal to one unit of length, describe a circle and draw two diameters AA' and BB' perpendicular to each other.

The circle with radius equal to 1 is called a unit circle, AA' the horizontal, and BB' the vertical diameter.

Let AOP be an acute angle,
We may

and let its value (in degrees, etc.) be denoted by x. regard the ≤ x as generated by a radius OP that revolves about / from the position OA to the position shown in the figure; viewed in this way, OP is called the moving radius.

Draw PM to OA, PN | OB. In the rt. A OPM the hypotenuse OP=1; therefore, sin x = PM; cos x = OM.

Since PM is equal to ON, and ON is the projection of OP on BB', and since OM is the projection of OP on AA', therefore, in a unit circle,

=

sin x = projection of moving radius on vertical diameter; cosa projection of moving radius on horizontal diameter. Through A and B draw tangents to the circle meeting OP, produced in T and S, respectively; then, in the rt. ▲ OAT, the leg OA-1, and in the rt. ▲ OBS, the leg OB=1; while the OSB=x. Therefore,

tan x=AT;
secx=OT;

cot x=
ᏣᏚᏟ X ---

BS;
OS;

vers x= AM; covers xBN.

These eight line values (as they may be termed) of the functions are all expressed in terms of the radius of the circle as a unit; and it is clear that as the angle varies in value the

line values of the functions will always remain equal numerically to the ratio values. Hence, in studying the changes in the functions as the angle is supposed to vary, we may employ the simpler line values instead of the ratio values.

EXERCISE III.

1. Represent by lines the functions of a larger angle than that shown in Fig. 3.

If x is an acute angle, show that

2. sinx is less than tan x.

3. secx is greater than tan x.

4. cscx is greater than cotx.

Construct the angle x if,

11. Show that the sine of an angle is equal to one-half the

chord of twice the angle.

12. Find x if sinx is equal to one-half the side of a regular inscribed decagon.

13. Given x and y, x+y being less than 90°; construct the value of sin (x + y) − sin x.

14. Given x and y, x+y being less than 90°; construct the value of tan (x + y) − sin (x + y) + tanx - sin x.

Given an angle x; construct an angle y such that, 15. siny 2 sinx.

17. tany 3 tan x.

19. Show by construction that 2 sin A> sin 2 A.

20. Given two angles A and B, A+B being less than 90°; show that sin (A+B) < (sin A+ sin B).

21. Given sin x in a unit circle; find the length of a line corresponding in position to sin x in a circle whose radius is r. 22. In a right triangle, given the hypotenuse c, and also sin =m, cos A= n; find the legs.

§ 4. CHANGES IN THE FUNCTIONS AS THE ANGLE CHANGES.

If we suppose the ally by the revolution

B

S" S

P

P

TH

T'

PAT

MM'M A
FIG. 4.

S

AOP, or x (Fig. 4) to increase graduof the moving radius OP about 0, the point P will move along the arc AB towards B, T will move along the tangent AT away from A, S will move along the tangent BS towards B, and M will move along the radius OA towards O.

Hence, the lines PM, AT, OT will gradually increase in length, and the lines OM, BS, OS will gradually decrease. That is,

As an acute angle increases, its sine, tangent, and secant also increase, while its cosine, cotangent, and cosecant decrease.

On the other hand, if we suppose x to decrease gradually, the reverse changes in its functions will occur.

If we suppose x to decrease to 0°, OP will coincide with OA and be parallel to BS. Therefore, PM and AT will vanish, OM will become equal to OA, while BS and OS will each be infinitely long, and be represented in value by the symbol oo.

And if we suppose x to increase to 90°, OP will coincide with OB and be parallel to AT. Therefore, PM and OS will each be equal to OB, OM and BS will vanish, while AT and OT will each be infinite in length.

Hence, as the angle x increases from 0° to 90°,

sinx increases from 0 to 1,

cosx decreases from 1 to 0,

tan x increases from 0 to ∞,

cotx decreases from ∞ to 0,
secx increases from 1 to co,
cscx decreases from co to 1.

The values of the functions of 0° and of 90° are the limiting values of the functions of an acute angle. It is evident that (disregarding the limiting values),

Sines and cosines are always less than 1;

Secants and cosecants are always greater than 1;

Tangents and cotangents have all values between 0 and ∞. REMARK. We are now able to understand why the sine, cosine, etc., of an angle are called functions of the angle. By a function of any magnitude is meant another magnitude which remains the same so long as the first magnitude remains the same, but changes in value for every change in the value of the first magnitude. This, as we now see, is the relation in which the sine, cosine, etc., of an angle stand to the angle.

§ 5. FUNCTIONS OF COMPLEMENTARY ANGLES. The general form of two complementary angles is A and 90° — A.

Each function of an acute angle is equal to the co-named function of the complementary angle.

NOTE. Cosine, cotangent, and cosecant are sometimes called cofunctions; the words are simply abbreviated forms of complement's sine, complement's tangent, and complement's secant.

Hence, also,

Any function of an angle between 45° and 90° may be found by taking the co-named function of the complementary angle between 0° and 45°.

EXERCISE IV.

1. Express the following functions as functions of the complementary angle:

2. Express the following functions as functions of an angle less than 45°:

§ 6.

RELATIONS OF THE FUNCTIONS OF AN ANGLE. Formula [1]. Since (Fig. 5) a2+b2= c2, therefore,

2

Therefore (§ 2), (sin A)2 + (cos A)2 = 1;

or, as usually written for convenience,

That is: The sum of the squares of the sine and the cosine of

an angle is equal to unity.