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CHAPTER IV.

GENERAL FORMULAS.

38. THE solution of oblique triangles requires the introduction of several trigonometrical formulas, which it is convenient to bring together and investigate all at once.

39. Problem. To find the sine of the sum of two angles. Solution. Let the two angles be BAC and B'AC (fig. 9), represented by the letters M and N. At any point C, in the line AC, erect the perpendicular BB'. From B let fall on AB' the perpendicular BP. Then represent the several lines as follows,

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Now the triangles BPB' and B'AC, being right-angled, and

having the angle B' common, are equiangular and similar.

Whence we derive the proportion

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The second member of this equation may be separated into

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whence, by substitution, we obtain

sin. (M + N) = sin. M cos. N + cos. M sin. N.

(33)

40. Problem. To find the sine of the difference of two angles.

Solution. Let the two angles be BAC and B'AC (fig. 10), repre-. sented by M and N. At any point C in the line AC erect the From B let fall on AB' the perpendicular BP. Then, using the notation of § 39, we have

perpendicular BB'C.

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The triangles B'AC and BB'P are similar, because they are right

angled, and the angles at B' are vertical and equal.

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41. Problem. To find the cosine of the sum of two angles. Solution. Making use of (fig. 9), with the notation of $39, and also the following

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The similar triangles BPB' and B'AC, give the proportion

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h/2

h12 — a12 = (AB')2 — (B'C)2 = (AC)2 = b2;

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42. Problem. To find the cosine of the difference of two angles.

Solution. Making use of (fig. 10) with the notation of the preceding section, we have

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The similar triangles BB'P and B'AC give the proportion.

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cos. (M — N) = cos. M cos. N+ sin. M sin. N.

(36)

43. Corollary. The similarity, in all but the signs, of the formulas (33) and (34) is such, that they may both be written in the same form, as follows,

sin. (M±N) = sin. M cos. N± cos. M sin. N,

(37)

in which the upper signs correspond with each other, and also the lower ones.

In the same way, by the comparison of (35) and (36), we are led to the form

cos. (M±N) = cos. M cos. N sin. M sin. N,

(38)

in which the upper signs correspond with each other, and also the

lower ones.

44. Corollary. The sum of the equations (33) and (34) is

sin. (M+N) + sin. (M — N) = 2 sin. M cos. N. Their difference is

(39)

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sin. (MN) sin. (MN) = 2 cos. M sin. N.

(40)

45. Corollary. The sum of (35) and (36) is

cos. (M + N) + cos. (M — N) =2 cos. M cos. N.

(41)

Their difference is

-

cos. (MN) cos. (M+N) = 2 sin. M sin. N.

(42)

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sin. A + sin. B = 2 sin. ≥ (A + B) cos. † (A — B)
sin. A sin. B2 cos. ≥ (A + B) sin. † (A — B)

(43)

(44)

cos. A cos. B = 2 cos. 1⁄2 (A + B) cos. ¿ (A
+
¿ — B)

(45)

1

cos. B — cos. A = 2 sin. § (A + B) sin. § (A — B). (46)

47. Corollary. The quotient, obtained by dividing (43) by (44), is

sin. A - sin. B
sin. A
sin. B

1 ≥
sin. (A+B) cos. (A — B)

cos. ¿ (A + B) sin. ¿ (A — B)

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