Page:EB1911 - Volume 27.djvu/288

an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality.

3. If a point moves from a position A to another position B on a straight line, it has described a length AB of the straight line. It Sm of is convenient to have a simple mode of indicating in portions of which direction on the straight line the length AB has an Infinite been described; this may be done by supposing that a Straight point moving in one specified direction 15 describing Lina a positive length, and when moving in the opposite direction a negative length. Thus, if a point moving from A to B is moving in the positive direction, we consider the length AB as positive; and, since a point moving from B to A is moving in the negative direction, we consider the length BA as negative. Hence any portion of an infinite straight line is considered to be positive or negative according to the direction in which we suppose this portion to be described by a moving point; which direction is the positive one is, of course, a matter of convention.

If perpendiculars AL, BM be drawn from two points, A, B on any straight line, not necessarily in the same plane with AB, the length LM, taken with the positive or negative sign according to the convention as stated above, is called the projection of AB on the given straight line; the each other projection of BA being ML has the oppositeProjections of Straight Lines
on each other.sign to the projection of AB. If two points A, B be joined by a number of lines in any manner, the algebraical sum of the projections of all these lines is LM-that is, the same as the projection of AB. Hence the sum of the projections of all the sides, taken in order, of any closed polygon, not necessarily plane, on any straight line, is zero. This principle of projections we shall apply below to obtain some of the most important propositions in trigonometry.

4. Let us now return to the conception of the generation of an angle as in fig. 1. Draw BOB′ at right angles to and equal to AA′. We shall suppose that the direction from A′ to A is the positive one for the straight line AOA′, and that from B′ to B for BOB′. Suppose OP of fixed length, equal to OA, and let PM, PN be drawn perpendicular to A′A, B′B respectively: then OM and ON, taken with their proper signs, are the projections of OP on A′A and B′B. The ratio of the projection of OP on B′B to the A absolute length of OP is dependent only on the magnitude of the angle POA, and is called the sine of that angle; the ratio of the projection of OP on A′A to the length OP is called the cosine of the angle POA. The ratio of the sine of an angle to its cosine is called the tangent of the angle, and that of the cosine to the sine the cotangent of the angle; the reciprocal of the cosine is called the secant, and that of sine the cosecant of the angle. These functions of an angle of magnitude α are denoted by sin α, cos α, tan α, cot α, sec α, cosec α respectively. If any straight line RS be drawn parallel to OP, the projection of RS on either of the straight lines A′A, B′B can be easily seen to bear to RS the same ratios which the corresponding projections of OP bear to OP; thus, if a be the angle which RS makes with A′A, the projections of RS on A′A, B′B are RS cos α. and RS sin a. respectively, where RS denotes the absolute length RS. It must be observed that the line SR is to be considered as parallel not to OP but to OP″, and therefore makes an angle π+α with A′A; this is consistent with the fact that the projections of SR are of opposite sign to those of RS. By observing the signs of the projections of OP for the positions P, P′, P″, P‴ of P we see that the sine and cosine of the angle POA are both positive; the sine of the angle P′OA is positive and its cosine is negative; both the sine and the cosine of the angle P″OA are negative; and the sine of the angle P‴OA is negative and its cosine positive. If α be the numerical value of the smallest angle of which OP and OA are boundaries, we see that, since these straight lines also bound all the angles 2nπ+α, where n is any positive or negative integer, the sines and cosines of all these angles are the same as the sine and cosine of a.. Hence the sine of any angle 2nπ+α is positive if α is between 0 and π and negative if α is between π and 2π, and the cosine of the same angle is positive if α is between 0 and 1/2π or 3/2π and 2π and negative if α is between 1/2π and 3/2π

Fig. 2.

Fig. 2.

In fig. 2 the angle POA is α, the angle P‴OA is -o., P'OA is π−α., P″A is π+α, POB is évr-a. By observing the signs of the projections we see that

sin(−α) = −sin α, sin(π−α), sin α, sin (π+α)-= −sin α,
cos(−α) =cos α, cos(1'r−α)= −cos α, cos(1r+α) = −cos α,
sin(1/2π−α) =cos (1, cos(π'l""(1)= sin α.
Also sin(1/2π+α) =sin(1r-Qvr−α) = sin(é1r−α)= cos α,
cos(1/2π+α) = −cos(π−1/2π−α) = −cos(1/2π−α) = −sin α.

From these equations we have tan(-a)=-tan a, tan(1r-a)= ~tan (l, t21fl(7f+U.)=*ft1l] a, tan(%1r-o.)=C0t a, tan (%'ll"j“U.>= -cot a, with corresponding equations for the cotangent.

The only angles for which the rejection of OP on B′B is the same as for the given angle POA (=α) are the two sets of angles bounded by OP, OA and OP', OA; these angles are 2nπ+α. and 2nπ+(π−α), and are all included in the formula nr-l-(-1)'a, where 1 is any integer; this therefore is the formula for all angles having the same sine as α. The only angles which have the sante cosine as a are those bounded by OA, OP and OA, OP′”, and these are all included in the formula 2nπ±α. Similarly it can be shown that nπ±α includes all the angles which have the same tangent as α.

From the Pythagorean theorem, the sum of the squares of the projections of any straight line upon two straight lines at right angles to one another is equal to the square on the projected line, we get sin²α+cos²α=1, and from this by the help of the definitions of the other functions we deduce the relations 1 + tan²α = sec²α, I + cot²α = cosec²a. We have now six relations between the six functions; (these enable us to express any five of these functions in terms of the sixth. The following table shows the values of the trigonometrical functions of the angles 0, 1/2π, π, 3/2π, 2π, and the signs of the functions of angles between these values; I denotes numerical increase and D numerical decrease:—

Angle	0	0...1/2π	1/2π	1/2π...π	π	π...3/2π	3/2π	3/2π...2π	2π
Sine	0	+I	1	+D	0	−I	−1	−D	0
Cosine	1	+D	0	−I	−1	−D	0	+I	1
Tangent	0	+I	±∞	−D	0	+I	±∞	−D	0
Cotangent	±∞	+D	0	−I	±∞	+D	0	−I	±∞
Secant	1	+I	±∞	-D	−I	−I	±∞	+D	1
Cosecant	±∞	+D	1	+I	±∞	-D	−1	−I	±∞

The correctness of the table may be verified from the figure by considering the magnitudes of the projections of OP for different positions.

The following table shows the sine and cosine of some angles for which the values of the functions may be obtained geometrically:—

sine cosine I,

π/12 15° /6;/2 V6-L-f'2 750 i§§ 1r

π/10 18° 'ir-1 v"50+2vs ...Q 2.

π/6 30° 2 .7 60 371

π/5 36° — . 5 ° 2.

π/4 45° V? Y 2 45° 3 1

4 cosine sine 4 p

These are obtained as follows. (1) 1/4π. The sine and cosine of this angle are equal to one another, since sin in-=cos (évr-iw); and since the sum of the squares of the sine and cosine is unity each is 1/√2. (2) 1/4π and 1/3π. Consider an equilateral triangle; the projection of one side on another is obviously half a side; hence the cosine of an angle of the triangle is 1/2 or cos 1/2π=1/2, and from this the sine is found. (3) π/10, π/5, 2π/5, 3π/10. In the triangle constructed in Euc. iv. 10 each angle at the base is 2/5π, and the vertical angle is 1/5π. If a be a side and b the base, we have by the construction a(a−b)=b²; hence 2b=a (1/5-I); the sine of -/r/Io is b/za or § (/5-1), and cos gr is a/2b=§ (/5-{-1) (4) fyr, ,521r. Consider a right-angled triangle, having an angle § -ir. Bisect this angle, then the opposite side is cut by the bisector in the ratio of /3 to 2; hence the length of the smaller segment is to that of the whole in the ratio of x/3 to x/3-1-2, therefore tan,131r={w/3/(/3-l-2)} tan évr or tan 1/12π=2−√3, and from this we can obtain sin, lsr and cos #svn

Fig. 3.

Fig. 3.

5. Draw a straight line OD making any angle A with a fixed straight line OA, and draw OF making D an angle B with OD, this angle being measured positively in the same direction as A; draw FE a perpendicular on DO (produced if necessary). The projection of OF on OA is the sum ofFormulae for Sine and Cosine of Sum and Difference of Two Angles. the projections of OE and EF on OA. Now OE is the projection of OF on DO, and is therefore equal to OF cos B, and EF is the projection of OF