CHAPTER III.

TRIGONOMETRIC FUNCTIONS OF TWO ANGLES.

42. Fundamental Formulæ. We now proceed to express the trigonometric functions of the sum and difference of two angles in terms of the trigonometric functions of the angles themselves.

The fundamental formulæ first to be established are the following:

NOTE.

Here x and y are angles; so that (x + y) and (x − y) are also angles. Hence, sin (x + y) is the sine of an angle, and is not the same as sin x + sin y. Sin (x + y) is a single fraction.

Sin x + sin y is the sum of two fractions.

43. To prove that

and

sin (x+y)= sin x cos y + cos x sin y,

cos (x + y): = cos x cos y — sin x sin y. Let the angle AOB = x, and the angle BOC = y; then the angle AOC=x+y.

In OC, the bounding line of the angle (x+y), take any point P, and draw PD, PE, perpendicular to OA and OB, respectively; draw EH, EK, perpendicular to PD and OA.

X

NOTE.These two formulæ have been obtained by a construction in which x + y is an acute angle; but the proof is perfectly general, and applies to angles of any magnitude whatever, by paying due regard to the algebraic signs. For example,

Let AOB = x, as before, and BOC= y; then AOC, measured in the positive direction, is the angle x + y.

In OC, take any point P, and draw PD, PE, perpendicular to OA and OB produced; draw EH and EK perpendicular to PD and OA.

B

EK-PH
OP

EK, PH
+
ОР ОР

K D

H

ОР

EK

=

==

ОЕ ОР PE OP

- sin (180°-x) cos (y-180°) + cos (180° - x) sin (y - 180°)

* The introduced line OE is the only line in the figure which is at once a side of two right triangles (OEK and OEP) into which EK and OP enter. A similar remark applies to PE.

= cos (180° - x) cos (y – 180°) + sin (180° — x) sin (y — 180°)

= cos x cos y - sin x sin y.

(Art. 35)

The student should notice that the words of the two proofs are very nearly the

Then the angle EPH = 90° – HEP = BEH = AOB

=x.

=

OK + EH OK EH
= +

ОР

ОР OP

= cos x cos y + sin x siny.

NOTE 1. -The sign in the expression of the sine is the same as it is in the angle expanded; in the cosine it is the opposite.

* P is taken in the line bounding the angle under consideration; i.e., AOC.

NOTE 2. In this proof the angle x − y is acute; but the proof, like the one given in Art. 43, applies to angles of any magnitude whatever. For example,

Let AOB, measured in the positive direction, = x, and BOC=y. Then AOC=x-y.

In OC take any point P, and draw C PD, PE, perpendicular to OA and OB produced: draw EH, EK, perpendicular to DP and AO produced.

Then,

= EOK = AOB = 360° — x,

angle EPH

and

= sin (360° - x) cos (180° — y) — cos (360° — x) sin (180° — y)

OK OE HE PE

OE OP PE OP

cos (360° - x) cos (180° — y) — sin (360° — x) sin (180° − y)

NOTE 3. The four fundamental formulæ just proved are very important, and must be committed to memory. It will be convenient to refer to them as the 'x, y' formulæ. From any one of them, all the others can be deduced in the following

manner:

Thus, from cos (x − y) to deduce sin (x+y). We have

Substitute 90° — x for x in (1), and it becomes

(1)

cos {90°-(x + y)}= cos (90° - x) cos y + sin (90° - x) siny.

(Art. 29)

.. sin(x+y)= sin x cos y + cos x sin y. The student should make the substitutions indicated below, and satisfy himself that the corresponding results follow:

From

sin (x+y) to deduce cos (x + y) substitute (90°+x) for x.