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Extension of formule to all angles.

333. (1) To extend the A + B formulæ to all positive acute values of A and B.

Let A and B be positive acute angles; but A + B obtuse.
Let A = 90° – C; and B = 90° – D.
Then A + B = 180° – (C + D), :. C + D is acute.
Now sin (A + B) i.e. sin {180° – (C+D)}

=sin (C+D), by Art. 308,
= sin ( cos D + cos C sin D, by Art. 329,
= cos (90° – C) sin (90° D) + sin (90° – C) cos (90° D),

by Art. 76,
i.e. cos A sin B + sin A cos B.

Q. E. D.
And cos (A + B) i.e. cos {180° – (C+D)}

=-cos (C + D), by Art. 308,
=-cos C cos D + sin C sin D, by Art. 329,
=- sin (90° – C) sin (90° – D) + cos (90° - C) cos (90° – D),

by Art. 76.
i.e. - sin A sin B + cos A cos B.

Q. E. D.

334. (2) To extend the A+B formule to all positive values of A and B.

Assume that for some particular two values of A and B,

sin (A + B) = sin A cos B + cos A sin B ... (1)

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

Then
sin (A + B') i.e. sin {A +(B+90°)} = cos (A + B), by Art. 326,
= cos A cos B – sin A sin B, by assumption,
= cos A sin (90° +B) + sin A cos (90° + B), by Art. 326,
i.e. cos A sin B' + sin A cos B'.

And

cos (A + B') i.e. cos {A + (B +90°)}=- sin (A + B), by Art. 326,

sin A cos B - cos A sin B, by assumption,

– sin A sin (90° +B) + cos A cos (90° + B), by Art. 326, i.e. - sin A sin B' + cos A cos B'.

Thus, assuming the formulæ (1) and (2) for any given values of A and B, we have proved them true for values A and B + 90°.

But they have been proved in last article to be true for all positive acute angles.

Hence they are true for all positive angles of any magnitude.

335. (3) To extend the A + B formulo to all values of A and B positive or negative.

Assuming the formulæ true for some particular two values of A and B, we may prove precisely as in the last article that they are true for A and B - 90°, by means of the formulæ of Art. 326,

sin (B - 90°) =- cos B and cos (B - 90°) = + sin B. Hence, by induction, the formulæ hold for all negative (as well as positive) values of A and B.

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336. (4) To prove the A - B formulæ from the A + B formule.

sin (A - B) i.e. sin {A + (- B)}
=sin A cos (B) + cos A sin (-B)
= sin A cos B cos A sin B by Art. 308,
cos (A - B) i.e. cos {A+ (-B)}
=cos A cos (- B) – sin A sin (- B)

= cos A cos B + sin A sin B by Art. 308. Hence the A B formulæ follow from the A + B formulæ, for any values of A and B.

337. Another method of extending the proof of the formulæ of Arts. 329, 330 to angles of any sign or magnitude is to substitute directed lengths for arithmetic lengths in those proofs.

The order of lettering has been so chosen that no change need be introduced in the general proofs, except that the lengths are to be regarded as directed lengths.

Now as far as the original angles A, B, A + B, A B are concerned, all the equations are seen by inspection to be true from the general definitions of ratios, because of the fundamental formula for directed lengths (LM) + (MN)= (LN).

But it is not evident by inspection that in all cases the following equations hold in sign as well as in magnitude, viz. (RP)

(RQ) = cos A and

= sin A. (QP)

(QP) The student may, by drawing appropriate figures, work out for himself some of the 64 cases which arise, according to the different values of A, B, A + B, A B.

But a more general proof will be given below.

338. In comparing the A - B with the A+B proofs the student should observe that since (RP)=-(PR) and so on, the proofs are exactly equivalent, except that, since in the (A + B) case, B=(YOZ) or (QOP); but in the A - B case, B=(ZOY) or (POQ), we have in the former,

(QP) sin B =

(OP)? in the latter,

(PQ) sin B =

(OP) In each case, of course sin (QOP) =

(QP)

(OP) and

sin (POQ) =

(PQ)

(OP)' The two cases are reduced to one case, if we always take (YOZ) to be B, and apply it to negative (as well as positive) values of B.

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$ 2.

PROJECTIONS AND ERECTIONS*. 339. DEF. 1.

DEF. 1. The projection of a directed length on an indefinite line is the directed distance between the two perpendiculars to the indefinite line drawn through the extremities of the directed length.

DEF. 2. The erection of a directed length on an indefinite line is the directed distance between the two parallels to the indefinite line drawn through the extremities of the directed length.

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Thus, if OH is any directed length, and X'X an indefinite line, and if OM, HBN be perpendiculars upon X'X, and if HQ, OBP be parallels to X'X (see figure) then

(MN) or (OB) is the projection of (OH) upon X'X; and
(PQ) or (BH) is the erection of (OH) upon X'X.

The projection is thus parallel to the indefinite line; and the erection is perpendicular to it.

It is clear that the erection upon one line is the projection upon any perpendicular to it; and vice versa.

* The definitions of these terms are given here on the supposition that all the lines, to which reference is made, are in one plane.

340. Prop. I. If A, B, C be any three points and X'X any line, the projection (or erection) of (AC) upon X'X is equal to the sum of the projections (or erections) of (AB) and (BC) upon X'X.

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For, let AL, BM, CN be perpendiculars upon X'X.
Then, however L, M, N may fall, we have

(LN) =(LM) + (MN), i.e. projection of (AC) = projection of (AB) + projection of (BC).

Since the erection is merely the projection on a perpendicular, the same proposition holds of erections.

341. Prop. II. The sum of the projections or erections of the sides of any closed figure, directed from point to point round the figure, is zero.

For, let ABCDEF be such a closed figure.
Then, by Prop. I.,

proj. of (AB) + proj. of (BC) = proj. of (AC)
and proj. of (AC) + proj. of (CD) = proj. of (AD)
and proj. of (AD) + proj. of (DE)= proj. of (AE)
and so on.
Finally

proj. of (AF) + proj. of (FA) = zero. :: proj. of (AB) + proj. of (BC) + ... + proj. of (FA) = zero. The same clearly holds of erections.

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