# Given the triangle ABC c=8, ∠b=30°, b=3 find all the other missing values Please show the complete solution with an explanation. This is the SSA case: the possibilities include no solution, exactly one solution, or two solutions. As sciencesolve's solution shows, there is no solution.

An alternative to the Law of Sines in this case is to use the Law of Cosines -- the advantage is in the case where there...

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This is the SSA case: the possibilities include no solution, exactly one solution, or two solutions. As sciencesolve's solution shows, there is no solution.

An alternative to the Law of Sines in this case is to use the Law of Cosines -- the advantage is in the case where there are two solutions, as both solutions will be provided.

Let BC=a. Then applying the Law of Cosines using angle B we get:

`b^2=a^2+c^2-(a)(c)cosB`

`3^2=a^2+8^2-(8)(a)cos30^(circ)`

`9=a^2+64-8a(sqrt(3)/2)`

`a^2-4sqrt(3)a+55=0`

Applying the quadratic formula we get:

`a=(4sqrt(3)+-sqrt(48-4(1)(55)))/2` . Since the discriminant is negative, there are no real solutions. Thus no triangle can be formed with the given attributes.

Approved by eNotes Editorial Team You may use law of sines to find the measure of angle C such that:

`b/(sin B) = c/(sin C)`

You need to substitute the values the problem provides such that:

`3/(sin 30^o) = 8/(sin C) =gt 3/(1/2) = 8/(sin C)`

`3*sin C = 8/2 =gt 3*sin C = 4 =gt sin C = 4/3`

Notice that evaluating `sin C ` yields `4/3`  that is larger than 1, which is impossible, hence, the problem does not provide informations being capable to be put in use to find the missing values.

Approved by eNotes Editorial Team