Imágenes de páginas
PDF
EPUB
[blocks in formation]

Introducing R, applying logarithms, and reducing, (4) becomes

log cos A[log 1p+log (3p—a)+a.c. log b+a.c. log c]. In like manner introduce R and apply logarithms to (5) and (6).

By subtracting both members of (1) from 1 and reducing we find

[blocks in formation]
[ocr errors]

( 1 p − b ) ( p −c)
p(p-a)

(≥ p − a) (≥ p—c)

(9) (6)=(12) tan Cp-a) (p—b).

pp-b)

Pp-c)

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

100. Problem.

To find the area of a triangle when two sides and their included angle are given.

Let k denote the area of the triangle ABC, of which the two sides. b and c and their included angle A are given.

(1) 2 kbp.

[blocks in formation]

b

B

a

[ocr errors]

Introducing R, and applying logarithms, we have log (2 k) = log blog clog sin A-10.

101. Examples.

C

1. Two sides of a triangle are 345.6 and 485, respectively, and their included angle is 38° 45' 40'; what is the area? Ans. 52468.

2. Two sides of a triangle are 784.25 and 1095.8, respectively, and their included angle is 85° 40′ 20′′; what is the area. Ans. 428470.

102. Problem.

To find the area of a triangle when the three sides are given.

By the last problem we find

(1) kbe sin A,

(2) sin A2 sin A cos A. Article 95, (5).

(3) sin A=

(p-b) p-c). Article 98, (7).

bc

(4)

(5) sin A

cos App―a). Article 98, (4).

bc

2 √ ≥ p (≥ p − a) (≥ p—b) (≥p—c}

bc

(6)

k = √1p(pa) (≥ p—b) (≥ p-c).

103. Examples.

1. The sides of a triangle are 40, 45, 55, required the area. Ans. 887.412.

2. The sides of a triangle are 467, 845, 756, required the area. Ans. 175508.

104. Problem.

Given the perimeter and angles of a triangle, required the sides.

b sin B

(1)

(2)

a sin A

C sin C a sin A

(3)

Adding and reducing by Articles 96, (5) and 95, (5),

we have

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Let p

(5)

a+b+c, and reduce by 96, (7), we have

p

a

[blocks in formation]

2 cos B cos C

sin A

psin A

cos B cos C

Introducing R and applying logarithms, we have

log a log p+ log sin A+

=

a. c. log cos B+ a. c. log cos C— 10.

Similar formulas can be found for b and c. But, after a is found, b and c can be more readily found by article 69.

[blocks in formation]

105. Examples.

150, A = 70°, B = 60°, C=50°, re

[blocks in formation]

2. Given p31234.36, A = 35° 45', B = 45° 28', C98° 47', required a, b, c.

3. Given p

Ans. a = 7985, b 9742.5, c

[blocks in formation]

375, A55° 46′ 18′′, B 82° 49′ 08′′,

C= 41° 24′ 34′′, required a, b, c.

Ans. a 125, b

=

150, c = 100.

106. Problem.

Given the three sides of a triangle, to find the radius of

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

.. (5) BOC+AOC+AOB= 1 (a + b + c) r = 1 pr.

But (6) ABC= √ ≥ p (≥ p − a) (p—b) (p—c).

S. N. 9.

.. (7) pr = VP(pa) (p—b) (p-c).

[merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small]

1. The three sides of a triangle are 20, 30, 40, respectively, required the radius of the inscribed circle. Ans. 6.455.

2. The three sides of a triangle are 100, 150, 200, respectively, required the radius of the inscribed circle. Ans. 32.275.

108. Problem.

Given the three sides of a triangle to find the radius of the circumscribed circle.

Let O be the center of the circle,

and R the radius.

Let OD be perpendicular to b, then

[merged small][merged small][merged small][ocr errors]
[blocks in formation]

The angle 0 the angle B, since each is measured

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

.'.

(3) R

=

4 k

41 p (pa) (p-b) (p-c)
v 1

Prove that the formula will be the same if the center is without the triangle.

« AnteriorContinuar »