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EXAMPLES.

1. Given a = 172, and A = 23°; find B, b, and c.

Ans. B = 670

b = 405.2062.

c = 440.2016. 2. Given a = 315, B = 60°; find a, b, and c.

Ans. A = 60°.

b = 545.59575.

630. 3. Given a = 2100, B= 57°; find A, b, and c.

Ans. A = 33o.

b = 3233.706,

3855.757

CASE IV.

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Given c and A, it is required to find a, b, and B. By equation (1),

a = c sin A. By equation (2),

b = c cos A,
B = 90° - A.

a =

b = 399.4295.

EXAMPLES. 1. Given c= 240, A = 35°; find B, a, and b.

Ans. B = 55°

137.6568.

b = 196.596. 2. Given c = 575, B = 44°; find A, a, and b.

Ans. A = 46°.

- 413.6205. 3. Given c = 7, A = 29°; find B, a, and b.

Ans. B = 60°.

a = 3.39367.

b = 6.12234. 3. The four cases computed by logarithmic tables.—The calculations in the examples already given are performed by means of Tables of Natural Sines. The examples which follow are intended as exercises in the use of Logarithmic Tables. The

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method of performing calculations of this kind is fully explained in the Appendix.

CASE I.

a

Given a and b, it is required to find A, B, and c. By equation (3),

tan A 2

ū Taking the logarithm of each side, log tan A - 10 = log a - log b,

a – log tan A

+ logo

- log b. To find c, we have by equation (1),

= IO

a

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Taking the logarithms of both sides,

log c = 10 + log a – log sin A.

c The value of A having been already found, we can, by means of this equation, find the value of log c; and then, by the tables, the value of c.

EXAMPLES. 1. Given a = 121, and b = 49,

10+ log 121 = 12.0827854

log 49 = 1.6901961

log tan A = 10.3925893 As this number is not found in the table of log tans, we proceed by Rule XII., Appendix, in order to find the corresponding angle :

log tan A = 10.3925893 log tan 67° 57' = 10.3925003

890 = diff. Tab. diff. = 3631.

890 x 60

14". 3631 Am. 1 = 67° 57' 14".

B = 22° 2' 46".

To find c,

10 + log 121 = 12.0827854 log sin 97° 57' = 9.9670125

2,1157729 Tab. diff. = 512. 512 X 14

119 (Rule XI.) 60

log c = 2.1157610 By applying Rule IV. we find,

Ans, c= 130.545. N. B.-In many cases it is more expeditious to find c from the equation, c= = V(a2 + b2), by the rule for the extraction of the square root, than by the preceding method. 2. Given a = 3, and b = 4; find c, 4, and B.

Ans. A = 36° 52' 11".

B = 53° 7' 49".

c= 5

3. Given a= 1 mile, and b = 3 furlongs, 7 perches; find the remaining side and angles.

Ans. A = 68° 21' 11".
B = 21°

49".

c = 1893.54 yds. 4. Given a = 144 feet, 6 inches, and b = a quarter of a mile;

a find A, B, and c.

Ans. A = 6° 14' 50".

B = 83° 45' 10".

c = 1327.8 ft. 5. Given a = 1341, and b= 1432 ; find A, B, and c.

Ans. A = 43° 7' 13".

B = 46° 52' 74".

C = 1961.87. 6. Given a = 1760, and b = 1000 ; find A, B, and c.

Ans. A = 60° 23' 44".

B = 29° 36' 16".

c= 2024.25. CASE II.

=

=

Given a and c, it is required to find b, A, and B. By equation (1),

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By means of this equation we can compute A. We find b by the equation,

62 = c? – Q2 = (c+a) (c-a). Taking the logarithms of both sides,

2 log b = log (c + a) + log (c - a).

EXAMPLES.

1. Given 4 = 13.2, and c = 127; find b, A, and B.

10 + log 13.2 = 11.1205739

log 127 = 2.1038037 log sin A = 9.0167702 log 5° 57' = 9.0156135

11567 = diff. Tab. diff. = 12104.

11567 x 60

= 57". 12104 Ans. A = 57' 57".

B = 84° 2' 3". To find by C+ a = 140,2; 6-a = 113.8

log 140.2

= 2.1467480 log 113.8 = 2.0561423

2)4.2028903

log b = 2.1014451 (Appendix, Rule IV.).

Ans. b = 126.3121. 2. Given a = 512, and e = 1007; find the angles A and B.

Ans. A = 30° 33' 36".

B = 59° 26' 24". 3. Given a = 32.712, and c= 96.2; find the angles A and B.

Ans. A = 19° 52' 46".

B = 70° 7' 14". 4. Given a = 123, and c = 157 ; find A, B, and b.

'Ans. A = 51° 34' 35".

B= 38° 25' 25". b= 97.5704.

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5. Given a = 576, and c = 880; find the angles A and B.

Ans. A = 40° 53' 7".

B = 49° 6' 53". 6. Given a = 21.7, and c = 54.31; find A, B, and b.

Ans. A = 23° 33' 2".

B = 66° 26' 58".

b = 49.7864.

CASE III. Given a and A, it is required to find B, b, and c. By equation (1),

a = c sin A; from which,

log G = 10 + log a – log sin A. By equation (3),

a = b tan A; from which,

log b = 10 + log a – log tan A.

EXAMPLES.

=

1. Given a = 13, and 4 = 35° 2'; find B, b, and c.

Ans. B = 54° 58'.

b= 18.543

= 22.646. 2. Given a = 1157, and B = 58° 3' 27"; find A, b, and c.

Ans. A = 31° 56' 33".

b= 1855.7290.

2186.8648. 3. Given a = 825, and B = 36°; find A, b, and c.

Ans. A = 54o.

b = 599.397

1019.756. 4. Given a = 1426, and A = 3° 21'; find B, b, and c.

Ans. B = 86° 39.

b= 24361.38

C = 24403.09 5. Given a = 28.75, and 4 = 17° 30' 30" ; find B, b, and c.

Ans. B = 72° 29' 30".

b= 91.1371.
C = 95.5643.

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