How do you determine whether $△ A B C$ $triangle ABC$ has no, one, or two solutions given $A = 95^{\circ}, a = 19, b = 12$ $A=95^circ, a=19, b=12$ ?

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How do you determine whether $△ A B C$ $triangle ABC$ has no, one, or two solutions given $A = 95^{\circ}, a = 19, b = 12$ $A=95^circ, a=19, b=12$ ?

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Answer 1 · 2017-06-10T09:04:49

one soltion only

$B = {39.0}^{0}, C = {46.0}^{0}, c = 13.7$ $B=39.0^0, C=46.0^0, c=13.7$

Explanation:

The triangle for this problem is:

NTS

enter image source here

The Sine rule

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a/(sinA)=b/(sinB)=c/(sinC)$

may be used ot find either a side OR an angle if one side and an opposite angle are know.

In this case we need to find an angle first

$\frac{19}{sin 95} = \frac{12}{sin B} = \frac{c}{sin C}$ $19/(sin95)=12/(sinB)=c/(sinC)$

we have to find $sin B$ $sinB$ first

$\frac{19}{sin 95} = \frac{12}{sin B}$ $19/(sin95)=12/(sinB)$

$\Rightarrow \frac{sin 95}{19} = \frac{sin B}{12}$ $=>sin95/19=sinB/12$

ie

$sin B = \frac{12 \times sin 95}{19}$ $sinB=(12xxsin95)/19$

$sin B = 0.6291755988$ $sinB=0.6291755988$

$\Rightarrow B = {38.98932587}^{0}$ $=>B=38.98932587^0$

$C = 180 - (95 + 38.98932587)$ $C=180-(95+38.98932587)$

$C = {46.01067413}^{0}$ $C=46.01067413^0$

to find the missing side

$\frac{c}{sin C} = \frac{19}{sin 95}$ $c/sinC=19/sin95$

$c = \frac{19 \times sin C}{sin 95}$ $c=(19xxsinC)/sin95$

#c=13.72213169

so the solution is , all given to 1dp

$B = {39.0}^{0}, C = {46.0}^{0}, c = 13.7$ $B=39.0^0, C=46.0^0, c=13.7$

with the sine rule there is always the possibility of a second set of solutions for the triangle--the AMBIGUOUS case.

to check we take the first angle found

$B = 39.9$ $B=39.9$

subtract from $180^0, because sin(180-x)=sinx$

$B'=180-39.9=140.1$

add this to A

$=95+140.1>180, :.$ no second solution

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How do you determine whether $△ A B C$ $triangle ABC$ has no, one, or two solutions given $A = 95^{\circ}, a = 19, b = 12$ $A=95^circ, a=19, b=12$ ?

1 Answer

Explanation:

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How do you determine whether triangle ABC△ABCtriangle ABC has no, one, or two solutions given A=95^circ, a=19, b=12A=95∘,a=19,b=12A=95^circ, a=19, b=12?

1 Answer

Explanation:

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How do you determine whether $△ A B C$ $triangle ABC$ has no, one, or two solutions given $A = 95^{\circ}, a = 19, b = 12$ $A=95^circ, a=19, b=12$ ?