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log Cosß= log Cos a + log Cos a – log Cos b:
then c=a +ß, under the same conditions as above.

Sixth Case: given A, B, a.

To determine 6:

sin B sinb=sin a ;

sin A

sin a

log Sin b=log Sin a + log Sin B – log Sin A.

sin A The above remarks on may here be applied to sin 6

sin B C, and c may be determined as above; or thus,

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log Sin ß=log Sin a + log Cos A – log Cos B : then C=a+ß, as above.

Assume tan a=tan a.cos B, log Tan a=log Tan a + log Cos B – logr.

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log Sin ß= log Sin a + log Cos A – log Cos B; then c=a+ß, as above. (W. Ch. xi; C. Ch. xviii; L. 221–62: Legendre, Trig. 84–91;

Delambre, Astron. 138-83.)

(27.) Series for the sine, cosine, &c. in terms of the arc.

23
25

27
sin x = 8
+

+ &c. a 1.2.3 1.2.3.4.5

1....7

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1

cosec =

+

8

+

+ &c.

1

7203 31x5 3.127x7
+
+

+
1.2.3 3.1....5 3.1....7 5.1....9
5.7.7309 1414477 211

+
3.1....11 3.5.7.1....13
2x

62 · 1382 211 tan x = x + +

+

+
3
3.5

32.5.7 34.5.7 34.52.7.11
21844x13

929569x15
+

+
36.52. 7.11.13 36.53.72.11.13

x3

17x7

+

E

+ &c.

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a

vers X=

1
2.25 ox?

2009
cot x =
3 32.5 33.5.7 33.52.7

3.5.7.11 1382 x11

4213

3617.x,15

- &c. 36.53.7.11.13 36.52. 7.11.13 37.54.7.11.13.17 ro

+

- &c.

B

(C. 266–302.) 1.2

1.2.3.4 1...6 (28.) Series expressing the inverse circular functions. 1 2 3 1.3 205

1.3.5 27 sin -18 = x +

+
2 3 2.45 2.4.6.7
203

@? tan -1x = x

3 5 7
1 1 1 1.3 1 1.3.5 1
+

+
2 303 2.4'50 2.4.6. 787
1

1.3 202 1.3.5 23
+

+
1.3.4 1.2' 5.42 1.2.3 7.43

+

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+ &c.

+

+ &c.

8

3

+

cosec

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+ &c.

vers -tx=(2x)}. {

1+

+ &c.

&c.}"

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(29.) Series for determining the value of 7.

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1 1 1 1
+

+
3 5

9

1 5.32

T

1

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6

7.33

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1

= tan4

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tan-1

43

7.1003 + &c.)

+

= 30° = {V3(1-378+

+ &c.

&c.) Euler's Method. - + tan-1

= (1-4+5-7+ &c)
+(1-5-6-7+ &e.).

1 Machin's Method. = 4 tan--

239 4

42

+
5 3.100 5.1002
1
1

1
(1

&c.). 239 3.57121 5.571212 n=3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 0938446, &c.

(Lacroix, Introd. Diff. Calc. 43–5.) (30.) Formulæ involving impossible quantities.

1 2T

(@v=i-8V=7). (exv1 +e-sv-1).

1 62rv-1-1 tan a =

Fi ev=+1
**v=1=cosa + J-1 sin x.
COS N X + -1 sin nx=(cos x + 1 sin x)".

m
1 sin x=(cosæ IV1 sin x)

n
1

{(cos x+/- 1 sin x)" – (cos x~~-1 sin a)"} 2 1 cos nx=}{(cos x+1 sin x)" + (cos æ— 1 sin x)"}.

sin x =

COS X =

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m COS

n

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sin nx=

If (a +61)8+av-1 = Cosc + ✓– 1 Sinc,

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a

.

and =tan x, then c=gx + 4h.log (a? + b*),

r=(a+ b*)£s.e=ht 1 1+ / - 1 tanc

log 2-1

1--1 tanc

(C. ch. viii; Hind, 247; L. 357–62.) log, 1=2(m-1)-1; m being any integer. Of these values only one is possible, which is obtained by putting m= =1.

(Lacr. Introd. 81.)

(31.) Formulæ for the sums of arcs, and multiple arcs.

Let the sum of the continued products of each combination of the sines of m of the arcs, aj, aq, &c. 0,9 into the cosines of the remaining n-m arcs, be represented by Cn-mSm; then

sin (a, + a2 + &c. + an)=Cn-Cn-2S, + C9-49.-&c.
cos (az + a, + &c. + an)=Cn-18,- Cn-38+ &c.

Let Tm be the sum of the continued products of each combination of m tangents,

T-T, + T; - &c. then tan (a, + a, + &c. +a)=

1-T, + T, - &c. If a,=Qq=&c. =an, then

n(n-1)(n-2) sin na=n(cos a)" – 1 sin a

(cos a)* -* (sin a)' +&c. 1. 2 3

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to }(n + 1) terms, if n is odd, and to in terms, if n is even.

n(n-1) cos na=(cos a)"

(cos a:)" – 2 (sin a)

1. 2
n(n-1)(n − 2)(n-3)
+

(cos a)" - * (sin a)* - &c. 2 3 4 to è (n + 1) terms, if n is odd, and to in +1 terms, if n is even.

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