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If b=90, c=70, and A=72^o48'30'', find B and C, given log 2=.3

Question

If $b = 90$ , $c = 70$ , and $A = 7 2^{o} 4 8^{'} 3 0^{''}$ , find B and C, given $l o g 2 = .30103$ , $L cot 3 6^{o} 2 4^{'} 1 5^{''} = 10.1323111$ ,
$L tan 9^{o} 3 7^{'} = 9.2290071$ ,
and $L tan 9^{o} 3 8^{'} = 9.2297735$ .

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Benjamin Discussed

b = 90

c = 70

, and

A = 7 2^{o} 4 8^{'} 3 0^{''}

, find B and C, given

l o g 2 = .30103

L cot 3 6^{o} 2 4^{'} 1 5^{''} = 10.1323111

L tan 9^{o} 3 7^{'} = 9.2290071

,
and

L tan 9^{o} 3 8^{'} = 9.2297735

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Text solutionVerified

tan \frac{B - C}{2} = \frac{b - c}{b + c} cot \frac{A}{2}

\Rightarrow

tan \frac{B - C}{2} = \frac{90 - 70}{90 + 70} cot \frac{7 2 ^{o} 4 8 ^{'} 33"}{2}

\Rightarrow

tan \frac{B - C}{2} = \frac{1}{8} = cot 3 6^{o} 2 4^{'} 15" = \frac{1}{2 ^{3}} cot 3 6^{o} 2 4^{'} 15"

L tan \frac{B - C}{2} = lo g 1 - 2 lo g 2 + L cot 3 6^{o} 2 4^{'} 15"

= 0 - 3 \times 0.30103 + 10.132311

L tan \frac{B - C}{2} = 9.2292211

Now,

L tan \frac{B - C}{2} - L tan 9^{o} 3 7^{'} = 9.2292211 - 9.2290071 = 0.000214

And also difference for

60" = 9.2297735 - 9.2290071 = 0.0007664

∴

Difference for

0.0002140 = \frac{0.0002140 \times 60}{0.0007664} = 17"

∴

\frac{B - C}{2} = 9^{o} 3 7^{'} 17"

---- ( 1 )

And

\frac{B + C}{2} = 9 0^{o} - \frac{a}{2} = 9 0^{o} - 3 6^{o} 2 4^{'} 15" = 5 3^{o} 3 5^{'} 4 5^{o}

--- ( 2 )

Solving equation ( 1 ) and ( 2 ) we get,

∠ B = 6 3^{o} 1 3^{'} 2"

and

∠ C = 4 3^{o} 5 8^{'} 28"

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Question Text	If $b = 90$ , $c = 70$ , and $A = 7 2^{o} 4 8^{'} 3 0^{''}$ , find B and C, given $l o g 2 = .30103$ , $L cot 3 6^{o} 2 4^{'} 1 5^{''} = 10.1323111$ , $L tan 9^{o} 3 7^{'} = 9.2290071$ , and $L tan 9^{o} 3 8^{'} = 9.2297735$ .
Answer Type	Text solution:1
Upvotes	150