Imágenes de páginas
PDF
EPUB

Continuing the reasoning, we see that the tangent of 270° is infinite, and that in the fourth quadrant the tangent is negative.

23 Changes in the secant. The secant is defined as the distance ON from 0 to the point N, in which the revolving side intersects the tangent line XN.

When m falls on X and the angle is zero, the secant is equal to OX, or unity. Hence

sec 0° = 1.

(3)

As m moves from X to Y, the secant increases without limit and becomes infinite when m reaches Y. Hence

[blocks in formation]

As m moves from Y through X' to Y', the intersection of the revolving line with the tangent line falls in the negative part of OM, or in the direction ON'. Hence:

In the second and third quadrants the secant is negative.

At Y', when the angle is 270°, the secant again becomes infinite.

Between Y and X, or in the fourth quadrant, it is again positive.

24. If we suppose the revolving line to make an integral number of revolutions from any point, it will return to its original position, and all the trigonometric functions will have the values corresponding to that position. Hence, if C is a circumference and n any integer,

In other words,

sin (nCa) = sin a;

tan (nC + a) = tan a ;

sec (nC + a) = sec a.

(5)

The values of the trigonometric functions are not altered by increasing the angle by any integral number of circumferences.

If the angle is increased indefinitely, the values of these functions continually repeat themselves. This fact is expressed by saying that these functions are periodic.

EXERCISES.

Prove the following expressions for the trigonometrical functions of angles of more than 90° by the necessary diagrams. The angle x may be supposed less than 90°, though this restriction is not necessary to the validity of the formulæ.

1. sin (90° + x) = sin (90° - x);

2. sin (180°+x) =

[ocr errors][merged small][merged small]
[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

NOTE. When we have the values of the trigonometric functions from 0 to 90°, we can by these formulæ find the values for all angles.

The Cosine, Cotangent, and Cosecant.

25. In the preceding sections we have supposed the side of the angle from which we count the degrees to go out toward the right, and the positive direction of motion to be opposite to that of the hands of a watch. But this restriction is only to fix the thought. We may suppose the angle to have any situation and to be counted in any direction without changing the values of the sine tangent, and secant, provided that we reckon the lengths of the lines representing the functions in the right way.

Let us count the angle from OY in the direction toward OX. The tangent line must then touch the circle at Y, and its positive direction must be toward the right.

[blocks in formation]

will have the same values as in an angle equal to POM counted in the usual way from OX toward Y.

Moreover, the changes of sign will be the same as before through the whole circle, namely:

From X to Y' (now the second quadrant) the sine PM will be positive because it is measured to the right.

From 'to X' (the third quadrant)

X

N

M

it will be negative because it is measured toward the left. It will also be negative from X' to Y (now the fourth quadrant). The corresponding propositions can be shown for the tangent and secant.

Now because angle XOM+angle MOY= angle XOY = 90°, MOY is the complement of XOM. Therefore the equations (a) may be written

PM sin comp. of XOM;

YN

=

tan comp. of XOM;

ON = sec comp. of XOM.

Because when we have an angle its complement can always be determined by subtracting it from 90°, we can always find the sine, tangent, and secant of the complement when we know the angle. Therefore the sine, tangent, and secant of the complement of an angle may be regarded as three additional trigonometrical functions of the angle itself. They are named thus:

The sine of the complement is called the cosine of the angle. The tangent of the complement is called the cotangent of the angle.

The secant of the complement is called the cosecant of the angle.

Thus the new functions are defined in the following way:

cosine a =

= sin (90° — a);

cotang a = tan (90° a);

(10)

cosecant a = sec (90° — a).

The words cosine, cotangent, and cosecant are abbreviated to

cos, cot, cosec, respectively.

The forms (10) enable us to find the cosine, cotangent, and cosecant of an angle when we know the sine, tangent, and secant of its complement. Thus if the cosine of 60° is required, we have cos 60° = sin (90° - 60°) = sin 30°.

Also, by substituting 90° - a for a, we find

[blocks in formation]

The versed-sine and co-versed-sine. Besides these six functions, two others, the versed-sine and co-versed-sine, are sometimes used. Their definitions are:

Versed-sine PX=1 cosine;

[ocr errors]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

we may take either OP or P'M as the cosine of XOM.

27. The general definitions of § 20 may be extended to com

[blocks in formation]

V. The cotangent of any angle XOM is represented by the length from Y to the point in which OM produced intersects YT.

VI. The cosecant of XOM is represented by the length of the side OM, intercepted between 0 and the line YT.

The algebraic signs of the several functions in the four quadrants are shown in the following diagram.

[blocks in formation]

28. The following are the limiting values of the trigonometrical functions:

I. Sine and cosine. The sine MP and cosine OP are necessarily not greater in absolute value than OM=1. The limits of these functions are therefore +1 and 1.

II. Secant and cosecant. Since the tangent line lies without the circle, a secant can never be less than unity in absolute magnitude. But we have found that it may increase to infinity in either the positive or negative direction. Hence the limits of the secant and cosecant are 1 and infinity, and 1 and

[ocr errors]
[ocr errors]

III. Tangent and cotangent. The limits of the tangent are easily seen to be ∞ and +∞, or a tangent and cotangent may have any value whatever.

29. When we know the numerical values of the sine, tangent, and secant of all angles from 0° to 90°, we have the values of all six functions of any angle whatever, because as we go around the circle the values of the functions are simply repetitions of the values between 0° and 90°.

Let a be any angle less than 90°. Then any angle in the first quadrant may be represented by a.

In the second quadrant it may be represented by 180° — α.

[ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
« AnteriorContinuar »