Question

how many triangles exist that fit the following criteria?
a = 65°, a = 8, b = 6

Explanation:

Step1: Apply the Law of Sines

We use the Law of Sines $\frac{a}{\sin A}=\frac{b}{\sin B}$. Substituting the given values $A = 65^{\circ},a = 8,b = 6$, we get $\sin B=\frac{b\sin A}{a}$.
So, $\sin B=\frac{6\sin65^{\circ}}{8}$.
Since $\sin65^{\circ}\approx0.9063$, then $\sin B=\frac{6\times0.9063}{8}=\frac{5.4378}{8}=0.6797$.

Step2: Find angle B

Since $\sin B = 0.6797$, then $B=\sin^{- 1}(0.6797)\approx43.1^{\circ}$ or $B = 180^{\circ}-43.1^{\circ}=136.9^{\circ}$.
But if $B = 136.9^{\circ}$, then $A + B=65^{\circ}+136.9^{\circ}=201.9^{\circ}>180^{\circ}$, which is not possible in a triangle.
So, there is only one valid value for $B$.

Step3: Determine the number of triangles

Since there is only one valid value for $B$, there is 1 triangle that satisfies the given criteria.

Answer:

1