Determine the number of solution for each possible triangle

B= 61 degrees a=12 b=8

The answer on the back of my book states zero.

Although I would think it has 2 because

12> b sin a when a is 29 degrees

Can anyone help me as I am slightly confused.

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- Dec 25th 2010, 01:27 PMhomeylova223Ambigious case for the law of sines?
Determine the number of solution for each possible triangle

B= 61 degrees a=12 b=8

The answer on the back of my book states zero.

Although I would think it has 2 because

12> b sin a when a is 29 degrees

Can anyone help me as I am slightly confused. - Dec 25th 2010, 01:43 PMArchie Meade
The triangle must obey the Sine Law, however..

$\displaystyle \displaystyle\frac{sin61^o}{8}\ \ne\ \frac{sin29^o}{12}$

$\displaystyle \displaystyle\frac{sinA}{a}=\frac{sinB}{b}\Rightar row\ sinA=\frac{12sin61^o}{8}=1.312$

which is not possible, since

$\displaystyle -1\ \le\ sinA\ \le\ 1$ - Dec 25th 2010, 01:44 PMdwsmith
Wikipedia offers and explanation and drawing which is helpful.

Law of sines - Wikipedia, the free encyclopedia - Dec 25th 2010, 08:09 PMSoroban
Hello, homeylova223!

Quote:

Determine the number of solution for this triangle:

. . $\displaystyle B= 61^o,\;a=12,\; b=8$

The answer on the back of my book states: zero..

If you had made a sketch, the answer would have been obvious.

Code:`C`

*

/:\

/ : \

a=12 / : \ b=8

/ : \

/ : \

/ :

/61o :

B * - - - + - - - * A

D

Draw altitude $\displaystyle \,CD$ to side $\displaystyle \,AB.$

In right triangle $\displaystyle CDB\!:\;\sin61^o \:=\:\dfrac{CD}{12} \quad\Rightarrow\quad CD \:=\:12\sin61^o \;\approx\;10.5$

Side $\displaystyle \,b$ is only 8 units long.

There istriangle.*no*