CHAPTER VI.

TRIGONOMETRIC RATIOS OF THE SUM AND

DIFFERENCE OF TWO ANGLES.

N.B. Another way of making the derivations shown in Arts. 46-48 is given in Note B of the Appendix. The method of projection, as it is called, used in Note B, is preferred by many.

46. Derivation of the sine and cosine of the sum of two angles when each of the angles is less than a right angle. In this article and the following one, careful regard must be paid to the direc tions in which lines and angles are measured, and to the order of the letters used in measuring them. See Arts. 36, 37,

40.

To deduce sin (A + B) and cos (A + B). Let A and B be two angles each of which is less than a right angle. Let the turning line revolve from the initial line OX, and about O describe the

angle XOL equal to A, and then revolve forward from the posi tion OL and describe the angle LOT equal to B. Thus, angle XOT=A+B. [In Fig. 45 a, A+B is less than 90°; in Fig. 45 b, A+B is greater than 90°.] Take any point P on OT, the terminal line of the angle (A + B), and draw PQ at right angles to OL, the terminal line of the angle A. From P, Q, draw PM, QN, at right angles to the initial line; and through Q draw VQR parallel to OX and intersecting MP in R.

Now, by definitions in Art. 40, and by Art. 45,

Now, by definitions in Art. 40, and by Art. 45,

1. Sin 75° = sin (30° + 45°) = sin 30° cos 45° + cos 30° sin 45°

2. Find cos 75° by putting 30° + 45° for 75° and using formula (2).

3. Deduce the sine and cosine of 15° from the results in Exs. 1, 2.

4. Find sin 90°, cos 90°, by putting 90° 30° + 60°.

90°

=

45° 45°. Also by putting 90° 75° + 15°.

Also by putting

5. Find sin 120°, cos 120°, by putting 120° = 60° + 60°; 120° = 90° + 30°; 120° 75° + 45°.

=

6. Find sin 150°, cos 150°, by putting 150° = 75° +75°;

7. Find sin 135°, cos 135°, by putting 135° 75° + 60°;

=

150° = 90° + 60°.

135° = 90° + 45°. 8. Given sin x = †, sin y = }, x and y both in the first quadrant; find sin (x + y), cos (x + y).

47. Derivation of the sine and cosine of the difference of two angles when each of the angles is less than a right angle. The construction and derivation are very similar to that made in the preceding article.

To deduce sin (A – B) and cos (A B). Let A and B be two angles each of which is less than a right angle, and let A be the greater. Let the turning line revolve from the initial line OX,

=

and about O describe the angle XOL equal to A, and then revolve backward from the position OL and describe the angle LOT equal to B. Then angle XOT= AB. Take any point P on OT, the terminal line of the angle (AB), and draw PQ at right angles to OL, the terminal line of the angle A. From P, Q, draw PM, QN, at right angles to the initial line; and through P draw RPV parallel to OX and intersecting NQ in R.

Now, by definitions in Art. 40, and by Art. 45,

:

sin VPQ = sin (180° – QPR) = sin QPR = cos RQP = cos A.

.. sin (4 – B) = sin A cos B-cos A sin B.

(3)

OP

OP

(Definitions, Art. 40.)

Now, by definitions in Art. 40, and by Art. 45,

=—cos QPR=-sin RQP=- sin A.

cos VPQ=cos (180° — QPR) =

..cos (AB) = cos A cos B + sin A sin B.

If B is greater than A, then the formula,

sin (BA) = sin B cos A cos B sin A,

can be deduced as above. Since

then

(4)

It is shown in Art. 48 that the formulas (1), (2), (3), (4), are true for all values of A and B. These formulas are called the addition and subtraction formulas or theorems in trigonometry. They are of such great importance, and so many thorems can be deduced by means of them, that they are called the fundamental formulas of trigonometry.* They should be memorized.

NOTE. Arts. 48, 49, may be omitted, if deemed advisable, until after the solution of triangles is completed. Art. 48 can also be shown geometrically.

* Adrian Romanus (1561-1625), professor of mathematics and medicine at the University of Louvain, was the first to prove the formula for sin(A+B). The formulas for cos(A ± B) and sin(A — B) were given by Pitiscus (1561-1613), a German mathematician and astronomer, in his Trigonometry published in 1595.

3. Find sin (x - y), cos (x − y) in the cases in Ex. 8, Art. 46.

48. Proof of addition and subtraction formulas for all values of A and B. These formulas have been proved in Arts. 46, 47, for values of A and B which are less than a right angle. In Art. 45 c it has been shown that for any angle, say X,

cos X = sin (90° + X), sin X=- cos (90° + X).

Hence, cos A= sin (90° + 4), sin A = cos (90°+ A),

cos (A+B)=sin(90°+A+B), sin (A+B)=—cos (90°+A+B). The substitution of these values for cos A, sin A, cos (A + B), sin (A + B), in (1), Art. 46, gives

cos (90°+A+B) = −cos (90°+A) cos B+sin (90°+A) sin B; ..cos (90°+A+B)=cos (90°+A) cos B-sin (90°+A) sin B. (1) The substitution of the same values in (2), Art. 46, gives

sin (90° + A + B)= sin (90°+ A) cos B+ cos (90° +A) sin B. (2)

Hence, formulas (1), (2), Art. 46, are true when one of the angles is increased by a right angle. In a similar way, these formulas can be shown to remain true when one of the angles in (1), (2), of this article is increased by a right angle. It is thus evident that the formulas are true, no matter how many right angles are added to either one or both of the angles. It can easily be shown that sin A=cos (4–90°), cos A=—sin (A—90°). Then, in the same way as that just employed, it can be shown that the formulas (1), (2), Art. 46, hold when either one or both of the angles is diminished by integral multiples of 90°. Hence, formulas (1), (2), Art. 46, are true for angles in any quadrant, that is, for all angles. In a similar way, formulas (3), (4), Art. 47, can be shown to be universally true.