49. Each fundamental formula contains the others. From any one of the four fundamental formulas, the remaining three can be derived. Thus for example: In (1) Art. 46, change A into 90° - A; then sin (90° — A + B)= sin (90° — A) cos B+ cos (90° — A) sin B. From this, sin (90° — A — B), i.e. cos (A — B)= cos A cos B + sin A sin B. In (1) Art. 46, change B into (— B); then sin (AB)= sin A cos (― B) + cos A sin (— B) = In (1) Art. 46, change A into (90° + A); then sin (90° + A + B) = sin (90° + A) cos B + cos (90° + A) sin B, whence, cos (A+B) = cos A cos B – sin A sin B. Ex. 1. From formula (2), Art. 46, derive the other three fundamental formulas. Ex. 2. So also, from formula (3), Art. 47. Ex. 3. So also, from formula (4), Art. 47. 50. Ratio of an angle in terms of the ratios of its half angle. In this article and Arts. 51, 52, a few deductions will be made from the addition and subtraction formulas, which have been shown to be true for all angles. These deductions are necessary for the explanations concerning triangles, as well as useful for other purposes. More ample opportunity will be afforded later for working exercises involving the use of these formulas. fundamental formulas may be brought together: sin (A+B) = sin A cos B + cos A sin B. The (1) (2) (3) (4) Let BA; then, from (1), that is, sin (A+A)= sin A cos A+ cos A sin A; sin 2 A= 2 sin A cos A. Similarly, from (3), cos 24 = cos2 A − sin2 4. Since cos2 A+ sin2 A = 1, it follows that and cos 2 A=1- 2 sin2 4. cos 2 A2 cos2 A-1. (5) (6) (7) (8) In formulas (1)-(8), A, B, denote any angles whatsoever. These formulas occur so often, and are so useful, that it is well to translate them into words. Thus, sine sum of any two angles = sin first cosine second sine difference of any two angles sine first cosine second - cosine first sine second cosine sum of any two angles = cosine first • cosine second - sine first · sine second cosine difference of any two angles cosine first cosine second A Since A is one-half of 2 A, formulas (5)–(8) can be translated as follows: sine any angle = 2 sine half-angle cosine half-angle, cosine any angle = (cosine half-angle)2 — (sine half-angle)2, .. cos2 221° = 1 (1 + 1 ) = 1 + √2 - 1+1.4142 = 2√2 2 x 1.4142 = .8536; ..cos 2210.9239. 7. Express cos 6 x, sin 6x, in terms of ratios of 3 x. hence, express sin 2 A, cos 2 A, in terms of sin A, cos A. 51. Tangents of the sum, and difference of two angles, and of twice an angle. Let A, B, be any two angles. It is required to find tan (A+B) and tan (A — B). On dividing each term of the numerator and the denominator of the second member by cos A cos B, there is obtained Formula (2) can also be deduced from (1) by changing B into Formulas (1), (2), (3), can be translated into words, as follows: 1. tan P=2, tan Q = }. Find tan (P+ Q), tan (P – Q). 52. Sums and differences of sines and cosines. The set of formulas (1)-(4), Art. 50, can be transformed into two other sets which are very useful. From (1), (2), (3), (4), Art. 50, on addition and subtraction, there is obtained: Substitution of these values of A, B, in (1)-(4) gives Formulas (1)-(4) with the members transposed, are useful for transforming products of sines and cosines into sums and differences; formulas (5) to (8) are useful for transforming sums and differences of sines and cosines into products. These formulas may be translated into words: Of any two angles, 2 sin one 2 cos one 2 cos one 2 sin one the sum of two sines = 2 sin half sum cos half difference, (5') the difference of two sines = 2 cos half sum · sin half difference, (6') the sum of two cosines = 2 cos half sum⚫ cos half difference, (7') the difference of two cosines = — - 2 sin half sum · sin half difference. (8') The difference between the first members of A, Art. 50, and C should be noted. N.B. Arts. 92–95 are similar in character to, and are merely a continuation of, Arts. 50-52. If deemed advisable, Arts. 91-95 can be taken up now. The student is advised to glance at them after solving the following exercises: |