If in these expressions, I-sa be substituted for c2, in the sines of the odd multiples of a, and in the cosines of the even multiples, the sines and cosines of such multiple arcs will be represented merely by the powers of the sine a. Sin 3a = 3s-453. (3.) Sin 5a = 5s-20s3+16s. Sin 7a = 7s-56s3+11255-6457. If the terms of the first table be repeatedly multiplied by 2s, and those of the second by 2c, observing the substitutions of cor. 2, there will result expressions for the sines and cosines. Thus, 2sin a2 =2s.s = cos 2a + 1, 4 sin a3 =-2s.cos 2a+2s = sin 3a + sin a + 2s = -sin 3a+3s, and 8 sin a4 =-2s.sin 3a + 2s.3s=+ cos 4a -cos 2a-3cos 2a+3=cos 4a-4cos2a+3. Again, 2 cos a =2c.c = cos 2a + 1, 4 cosa3 = 2c.cos 2a+2c= cos 3a + cos a + 2c = cos 3a + 3 cosa, and 8 cos a = 2c.cos 3a + 2c.3cosa cos4a+cos2a+3cos2a+3=cos4a+4cos2a+3. In this manner, the following tables are formed. 32 Sin a= 4 cos 2a + 3. 5 sin 3a + 10s. - cos 6a + 6 cos 4a - 15 cos 2a +10. 64 Sin a = - sin 7a +7 sin 5a-21sin 3a+35s. 32 Cos a = cos 6a + 6 costa + 15 cos 2a + 10. 64 Cos a' = cos 7a + 7 cos5a + 21 cos3a + 35c. &c. &c. &c. PROP. IV. THEOR. The sum of the sines of two arcs is to their difference, as the tangent of half the sum of those arcs to the tangent of half the difference. If A and B denote two arcs; sinA+sinB: sin A-sinB :: tan+B: tan-B 1 For, let AC and AC' be the sum aad difference of the is parallel to C'E', and CK to HL, CE: C'E':: CK: CK (VI. 2. El.) HL: H'L; and consequently CE+C'E' : CE-CE': : HL+HL: HL-HL', that is, 2BL: 2BH, or BL: BH. But CE and C'E' are the sines of the arcs AC and AC', and BL and BH are the tangents of AB and BC, or of half the sum and half the difference of those arcs. Wherefore sinAC+sinAC': sinACAC+AC sinAC':: tan 2 : tan AC-AC Cor. 1. The sines of the sum and difference of two arcs are proportional to the sum and difference of their tangents. For CE : C'E' : : HL, or BL+BH : HL, or BL-BH; that is, resuming the general notation, sin(A+B): sin(AB): : tanA+tanB : tanA-tanB. Cor. 2. Let the greater arc be equal to a quadrant; and R+sinB: R-sinB :: tan(45°+B): tan(45°-B), or cot(45°++B). But, the radius being a mean proportional between the tangent and cotangent of any arc, and the cosine of an arc being a mean proportional between the sum and difference of the radius and the sine, it follows that R+sinB: cosB :: R: tan(45°-B), and R-sin B: cosB, or cosB: R+sinB :: R: tan(45°+B). 1 Or, if instead of B, there be substituted its complement, these analogies will become R+cosB: SinB :: R: tanB, and R-cosB: sinB :: R: cotB. Cor. 3. Since cosB: R:: R-sinB: tan(45°-B), and cosB: R::R+sinB: tan(45°++B), therefore (VI. 19. EI.) COSB: R::2R: tan(45°+B)+tan(45°++B); that is, supposing B to be the complement of 2C, sin2C : 2R :: R: tanC+cotC. But (Prop. 1. cor. 1.) R.sin2C=2cosC. sinC, and consequently cosC.sinC : R2 : : R : tanC+cotC. Cor. 4. Since (4 cor. def.) cosB: R:: R: secB, and (3. cor. def.) cosB: sin B:: R: tanB, therefore cosB: R+sinB :: R : tanB+secB, and consequently (2. cor. def.) tan(45°++B)=tanB+secB.-This also appears clearly from the figure, on supposing OH'=H'L', or the angle LOH' equal to OLH', and consequently the arc AC' equal to the complement of AB. PROP. V. THEOR. As the difference of the square of the radius and the rectangle under the tangents of two arcs, is to the square of the radius, -so is the sum of their tangents, to the tangent of the sum of the arcs. Let A and B denote any two arcs; then, In reference to a diagram, let AB and BC be the two arcs, AD and BE their tangents, and AF consequently. the tangent of their sum HC. From the centre O, draw to meet the extension of this tangent, draw OH perpen or by alternation AG: AH :: DI: OH. Again, since the parallels DI and OH are intercepted by the diverging lines FH and FO, (VI. 2.) DI: OH :: FD: FH; wherefore AG: AH :: FD: FH, and (V. 10. El.) GH : AH :: DH : FH : : (V. 19. 1. cor. El.) DG: AF. Consequently (V. 25. cor. 2. El.) GH.AD: AH.AD::DG:AF; but (VI. 15. cor. El.) AH.AD=OA2, and hence GH,AD = OA AD.AG; wherefore OA-AD.BE: OA :: DG: AF. Now OA is the radius, AD and BE the tangents of the arcs AB and BC, DG their sum, and AF the tangent of the compound arc AC; consequently the proposition is manifest. Cor. 1. Hence it follows, by changing the position of the figure;-That as the sum of the square of the radius, and R |