(2) Sin 135° = sin (90° + 45o) = sin 90° cos 45°

(3) Sin (90° + A) = sin 90° cos A+ cos 90° sin A

(4) Sin (180° + A) = sin 180° cos A + cos 180° sin A

= 0.cos A - 1. sin A sin A:

=

==

cos (180° + A) = cos 180° cos A - sin 180° sin A

and the same kind of proceeding may be adopted in all quantities of this description.

39. If any one of the four fundamental formulæ established in (36) and (37) be given, the others may be deduced from it, by means of their known relations : thus, if we assume sin (A + B) = sin A cos B + cos A sin B, we shall have

sin (A-B) = sin {(180° - A) + B} = sin (180° - A) cos B + cos (180° – A) sin B = sin A cos B – cos A sin B : cos (A + B) = sin {(90° - A) - B} = sin (90° - A) cos B

cos (90° - A) sin B = cos A cos B - sin A sin B:

cos (A-B) = cos {(180° — A) + B} = cos(180° - A) cos B

+ sin (180o – A) sin B = cos A cos B + sin A sin B.

40. Since, sin (A + B) = sin A cos B + cos A sin B : sin (4 – B) = sin A cos B cos A sin B: cos (A + B) = cos A cos B - sin A sin B : cos (A-B) = cos A cos B + sin A sin B :

we have, by addition and subtraction,

sin (A + B) + sin (A – B) = 2 sin A cos B :
sin (A + B) – sin (A – B) = 2 cos A sin B :
cos (A – B) + cos (A + B) = 2 cos A cos B :

cos (A-B) - cos (A + B) = 2 sin A sin B :

and these formulæ will evidently enable us to express, in the forms of Sums or Differences, each of the Products in their latter members, and they deserve attention.

Ex. If A = 30°, we shall have

sin (30o + B) + sin (30° – B) = 2 sin 30o cos B = cos B : sin (30°+B) - sin (30° – B)=2 cos 30° sin B=√3 sin B: &c.

41. COR. From the last article, we have

sin (A + B) = 2 sin A cos B – sin (A – B),

cos (A + B) = cos (A - B) - 2 sin A sin B, &c :

which will serve to derive the functions of an angle from those of inferior angles connected with it: thus, if A = 30°, and B = 1';

sin 30° 1′ = cos 1′— sin 29° 59′ :

cos 30° 1'

= cos 29° 59′- sin l': &c.

that is, if the values of the sines and cosines of all angles up to 30° be known, those of all angles up to 60°, will

then be determined by subtraction only. By putting 45°, 60°, &c. for A or B, other useful results are obtained.

42. The operation of multiplication gives

sin (A + B) sin (A – B)

= (sin A cos B+cos A sin B) (sin A cos B- cos A sin B)

=

=

sin A cos B-cos A sin B

= sin'A (1-sin3B) - (1 - sin3A) sin B

= sin2A – sin2A sinoB - sin'B + sin2A sin'B

=

= sin3A sin3 B = (sin A + sin B) (sin A-sin B):

or, =

=

= cos2 B - cos2 A = (cos B + cos A) (cos B – cos A) : similarly, cos (A + B) cos (A - B) = 1 - sin'Asin3B

These formulæ may be used to deduce the sines and cosines of angles from those of inferior angles, by means of the operations of common Arithmetic: thus, if A = 2o, and B = 1o, we have

sin 30

=

(sin 90 + sin 1) (sin 20 – sin 1)

sin I

: &c.

43. By division of the numerator and denominator, by cos A cos B, we have

cos (A + B)

cot A cot B-1

=

=

similarly,

cos (A - B)

cot Atan B cot A cot B + 1 *

The object of these formulæ is to express the ratios of

the Sums and Differences of functions of the latter description, by means of single functions.

44. To express the sine and cosine of 2 A in terms of those of A.

By means of the general formulæ of Article (36):
sin 2A = sin (A + A)

45. COR. 1. From the last two results we have immediately,

2 sin A 1

=

cos 24, and 2 cos2A = 1 + cos 2A:

and by the substitution of A and A in the places of 24 and A respectively, we obtain

sin A = 2 sin cos A, 2 sin21⁄4A = 1 − cos A,

2 cos4= 1 + cos A,

which are formulæ of great utility in the subsequent parts of the work.

46.

COR. 2. We may easily express sin 2 A and cos 2 A in terms of tan A: for,

and these formulæ being rational, are frequently very convenient in practice.

47. COR. 3. Hence, sin 3A = sin (2A + A)

=

= 2 sin A (1-sin3A) + (1 - 2 sin3) sin A

(2 cos3A - 1) cos A 2 (1-cos2) cos A

two results which ought to be committed to memory.

48. To express the sine and cosine of A in terms of the sine of 2 A.

Since, 1 = cos2 A + sin2 A, and sin 2A we have, by addition and subtraction,

=

2 sin A cos A;

1 + sin 2A = cos2A + 2 sin A cos A + sin A

1 - sin 24 = cos3A – 2 sin A cos A + sin3A

= (cos Asin A):

whence, if cos A and sin A be positive quantities, and the square roots be extracted, we find

cos A - sin A =√1 - sin 24:

in the latter of which the upper or lower sign must evidently be used according as cos A is greater or less than sin A, ór, according as A is less or greater than 45o, by example (1) of Article (35):