93. In the last article, for the sake of simplicity we have supposed the angle A to be less than a right angle, but all the formulæ of this chapter may be shewn to be true for angles of any magnitude. A general proof of one case is given in Art. 102, and the same method may be applied to all the other cases.

94. If the angles are expressed in radian measure, the formulæ of Art. 92 become

Example 1. Find the sine and cosine of 120°.

95. DEFINITION. When the sum of two angles is equal to two right angles each is said to be the supplement of the other and the angles are said to be supplementary. Thus if A is any angle its supplement is 180° - A.

96. The results of Art. 92 are so important in a later part of the subject that it is desirable to emphasize them. We therefore repeat them in a verbal form:

the sines of supplementary angles are equal in magnitude and are of the same sign;

the cosines of supplementary angles are equal in magnitude b are of opposite sign;

the tangents of supplementary angles are equal in magni but are of opposite sign.

97. Circular Functions of 180°+A.

Take any straight line XOX' and let a radius vector starting from OX revolve until it has traced the angle A, taking up the position OP.

Again, let the radius vec

tor starting from OX revolve

through 180° into the position

M'

A

X'

M X

OX', and then further through an angle A, taking up the final position OP'. Thus XOP' is the angle 180°+4.

From P and P' draw PM and P'M' perpendicular to XX'; then OP and OP' are in the same straight line, and by Euc. 1. 26 the triangles OPM and OP'M' are geometrically equal.

and M'P' is equal to MP in magnitude but is of opposite sign;

and OM' is equal to OM in magnitude but is of opposite sign;

Expressed in radian measure, the above formulæ are written sin (+0)=sin 0, COS (+8)= cos 0, tan (+0)=tan 8.

In these results we may draw especial attention to the fact that an angle may be increased or diminished by two right angles as often as we please without altering the value of the tangent.

Example. Find the value of cot 210°.

cot 210° cot (180°+30°) = cot 30°=√3.

=

98. Circular Functions of 90°+A.

Take any straight line XOX', and let a radius vector starting from OX revolve until it has traced the angle A, taking up the position OP.

Again, let the radius vector starting from OX revolve through 90° into the position OY, and then further through X' an angle A, taking up the final

M' O

position OP'. Thus XOP' is the angle 90° + A.

then

A

M X

From P and P' draw PM and P'M' perpendicular to XX';
LM'P'OL P'OY=A= L POM.

By Euc. 1. 26, the triangles OPM and OP'M' are geometrically equal; hence

M'P' is equal to OM in magnitude and is of the same sign, and OM' is equal to MP in magnitude but is of opposite sign.

Example 2. Find the values of tan (270° +A) and cos

· (3 + 0).

tan (270° +4)=tan (180° +90°+4)=tan (90° +A) = − cot A ;

It is especially worthy of notice that we may change the sign of an angle without altering the value of its cosine.

Example. Find the values of

cosec (-210°)=

cosec (-210°) and cos (4 - 270°).

cosec 210°: = - cosec (180° +30°)=cosec 30°= 2.

cos (4 – 270°) = cos (270° -- A) = cos (180° + 90° – A)

cos (90° - A) = − sin A.

100. If ƒ (4) denotes a function of A which is unaltered in magnitude and sign when A is written for A, then f(A) is said to be an even function of A. In this case f(-— A)=ƒ(A).

If when A is written for A, the sign of ƒ (4) is changed while the magnitude remains unaltered, f(A) is said to be an odd function of A, and in this case ƒ (-A)= −ƒ(4).

From the last article it will be seen that

cos A and sec A are even functions of A,

sin A, cosec A, tan A, cot A are odd functions of A.

Express in the simplest form:

25. tan (180°+A) sin (90° + A) sec (90° - A).

26.

27.

cos (90°+4)+sin (180° – A) — sin (180° + A) – sin (– A). sec (180°+A) sec (180° – A)+cot (90° + A) tan (180°+A).