CASE III. A in the third quadrant, — A in the second.

Whence we obtain the formulæ (9) as before.

CASE IV. A in the fourth quadrant, —A in the first.

Whence we obtain the formulæ (9) as before.

43. The results of Art. 42 may be stated as follows:

The sine, cosine, tangent, cotangent, secant, and cosecant of a negative angle are equal respectively to the negative sine, the cosine, the negative tangent, the negative cotangent, the secant, and the negative cosecant, of the absolute value of the angle.

FUNCTIONS OF (90° + A) IN TERMS OF THOSE OF A. 44. To prove the formula, sin (90° +4)= cos A, tan (90° + A) = —cot A, sec(90°+4)=-esc A, for any value of A.

cos (90°+4)=sin A,
cot (90° + A) = −tan A, 》 (10)
csc (90° +4)=

sec A,

There will be four cases, according as A is in the first, second, third, or fourth quadrant.

In each figure, let the positive angle XOP (indicated by the full arc) represent the angle A, and the positive angle XOP' (indicated by the dotted arc) the angle (90° + A).

Lay off OP=OP', and draw PM and P'M' perpendicular to XX'.

Since OP and OM are perpendicular to OP' and P'M' respectively, the angles POM and OP'M' are equal.

Then the right triangles OPM and OP'M' have the hypotenuse OP and the angle POM of one equal to the hypotenuse OP' and the angle OP'M' of the other, and are equal.

Whence, PM = OM' and OM= P'M'.

Let PMOM' = a, OM = P'M' = b, and OP = OP' = c.

CASE I. A in the first quadrant, 90° + A in the second.'

It is evident from the above that the sine and cosecant of 90° + A are equal respectively to the cosine and secant of A, while the cosine, tangent, cotangent, and secant of 90° + A are equal respectively to the negatives of the sine, cotangent, tangent, and cosecant of A.

Whence we obtain the formulæ (10).

CASE II. A in the second quadrant, 90° + A in the third.

Whence we obtain the formulæ (10) as before.

CASE III. A in the third quadrant, 90° + A in the fourth.

Whence we obtain the formulæ (10) as before.

CASE IV. A in the fourth quadrant, 90° + A in the first.

Whence we obtain the formulæ (10) as before.

APPLICATION OF ALGEBRAIC SIGNS.

27

45. The results of Art. 44 may be stated in the following form:

The sine, cosine, tangent, cotangent, secant, and cosecant of any angle are equal respectively to the cosine, the negative sine, the negative cotangent, the negative tangent, the negative cosecant, and the secant, of an angle 90° less.

46. To find the values of the functions of 90° — A in terms of those of A.

We have thus proved the formulæ of Art. 14 for any value of A.

47. To find the values of the functions of 180° — A in terms of those of A.

Since the angles A and 180°- A are supplements of each other, these formulæ express the values of the functions of the supplement of an angle in terms of those of the angle itself.