Question

law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$
how many distinct triangles can be formed for which $mangle a = 75^{circ}, a = 2$, and $b = 3$?
no triangles can be formed.
one triangle can be formed where angle b is about 15°.
one triangle can be formed where angle b is about 40°.
two triangles can be formed where angle b is 40° or 140°.

Explanation:

Step1: Apply the law of sines

By the law of sines, $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}$. Substitute $A = 75^{\circ}$, $a = 2$, and $b = 3$ into the formula: $\sin(B)=\frac{b\sin(A)}{a}=\frac{3\sin(75^{\circ})}{2}$.
Since $\sin(75^{\circ})=\sin(45^{\circ}+ 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. Then $\sin(B)=\frac{3\times0.9659}{2}=1.44885$.

Step2: Analyze the value of $\sin(B)$

The range of the sine - function is $[- 1,1]$. Since $\sin(B)=1.44885>1$, no triangle can be formed.

Answer:

No triangles can be formed.