FUNCTIONS OF π +0.

THEOR. 9. If be any plane angle, then:

sin }π+0=+ cos 0, cos1⁄4π +0=— sin 0, tan1⁄2π+0=— cot 0, csc }π+0=+ sec 0, sec 1⁄2π +0= = — csc 0, cot π +0=—tan 0.

For, let xop be any plane angle 0, draw oq so that YOQ = XOP, and take oQ = OP ;

and

+0, and oq = op,

the abscissa and ordinate of P as to ox are equal to the abscissa and ordinate of Q as to or, and equal to the ordinate and the opposite of the abscissa of Q as to ox,

EXAMPLES.

1...6. In functions of positive angles less than a right angle express the values of :

6. csc 635°, 835°, -1035°, -1235°, 37π, -32π.

7...12. In functions of zero or of positive angles not greater than half a right angle express the values of:

17. A function of the sum or difference of

multiple of is as large as the

same
Co-

18. Prove vers (180 − a) + vers (360 — a) = 2.

and an

even

odd

function of 0.

19. Prove cos2 A + cos2 (90+ a) + cos2 (180 + a)

+cos2 (270+ A) = 2.

20. What values of x satisfy the equation sin 2 x = cos 3x?

§ 5. PROJECTIONS.

THEOR. 10. The x-projection of a limited straight line is the y-projection

product of the line by the {

cosine
sine

of its angle with the x-axis.

For, let ox, or be the axes, let TR be any line meeting ox in T and let PQ be any segment of TR;

draw AP, BQ perpendicular to ox, and CP, DQ perpendicular to or;

of a broken line is equal to the sum of

COR. The x-projection y-projection the products of the parts, each multiplied by the

angle with the x-axis.

EXAMPLE. Let 7, m be two lines, and a a segment of 1; project a on m, this projection on 7, this on m, and so on, and find the sum of all the projections on m.

§ 6. FUNCTIONS OF SUMS AND OF DIFFERENCES.

THEOR. 11. (Addition-theorem.) If 0, 0' be any two plane angles, sin 0 ± 0' = sin cos 0' ± cos 0 sin 0',

then

cos 00' = cos 0 cos 0' = sin 0 sin 0',

For, let XOP, POQ be any two plane angles 0, 0' ;

through P draw PR normal to OP, cutting ox, oq in H, K;

and

= sin cos 0'

and

OK sin 0 + 0' = OP sin 0 + PK sin 17 +0

= OP sin + PK COS 0,
OP=OK COS 0', PK OK sin 0',
.. OK sin + 0' = OK sin
... sin 00' = sin cos 0'+ cos 0 sin 0';
... sin 0' = sin 0 cos 0'+ cos 0 sin

So, ·.· OK COS @ +0' = OP COS 0 + PK COS π + 0

[constr. [th. 10, cr.

[th. 9 [df.

cos 0' + OK Cos sin ',

Q. E. D.

- O'

cos (0+0') cos (00')=-2 sin 0 sin O'.

The reader will note the complete generality of ths. 10, 11, and the consequent generality of all the theorems and corollaries that follow and depend upon them. Considering the great importance of th. 11, he should write out each of the four equations involved, so that it may stand out clearly by itself; and he should translate these equations into words and commit them to memory. This suggestion as to translation may apply to many other theorems that are given only in formula.

CONVERSION FORMULE.

THEOR. 12. If 0, 0' be any two plane angles, then :

sin (10+0+100') + sin (10+ 0' — 10 — 0')

cos

For

and