Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no
triangle at all. Solve any triangle(s) that results.
-$\mathrm { a } = 14 , \quad \mathrm {~b} = 12 , \quad \beta = 25 ^ { \circ }$
A) one triangle
$$\alpha = 150.46 ^ { \circ } , \gamma = 4.54 ^ { \circ } , \mathrm { c } = 2.25$$

B) one triangle
$$\alpha = 29.54 ^ { \circ } , \gamma = 125.46 ^ { \circ } , \mathrm { c } = 23.13$$
C) two triangles
$$\begin{array} { l }
\alpha _ { 1 } = 29.54 ^ { \circ } , \gamma _ { 1 } = 125.46 ^ { \circ } , \mathrm { c } _ { 1 } = 23.13 \text { or } \
\alpha _ { 2 } = 150.46 ^ { \circ } , \gamma _ { 2 } = 4.54 ^ { \circ } , \mathrm { c } _ { 2 } = 2.25
\end{array}$$
D) no triangle

Question

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no
triangle at all. Solve any triangle(s) that results.
-$\mathrm { a } = 14 , \quad \mathrm {~b} = 12 , \quad \beta = 25 ^ { \circ }$
A) one triangle
$$\alpha = 150.46 ^ { \circ } , \gamma = 4.54 ^ { \circ } , \mathrm { c } = 2.25$$

B) one triangle
$$\alpha = 29.54 ^ { \circ } , \gamma = 125.46 ^ { \circ } , \mathrm { c } = 23.13$$
C) two triangles
$$\begin{array} { l } 
\alpha _ { 1 } = 29.54 ^ { \circ } , \gamma _ { 1 } = 125.46 ^ { \circ } , \mathrm { c } _ { 1 } = 23.13 \text { or } \
\alpha _ { 2 } = 150.46 ^ { \circ } , \gamma _ { 2 } = 4.54 ^ { \circ } , \mathrm { c } _ { 2 } = 2.25
\end{array}$$
D) no triangle

Quizplus · Accepted Answer

Given $a = 14$, $b = 12$, and $\beta = 25^\circ$, we use the Law of Sines to determine the possible values of $\alpha$. Since $\sin(\beta)/b = \sin(\alpha)/a$, there can be two different angles $\alpha$ that satisfy this equation due to the Sine function's properties in different quadrants (acute and obtuse angles). This can lead to two different triangles, as shown in option C, where both sets of angles and the corresponding side lengths are valid solutions.

Two Sides and an Angle Are Given a=14, b=12,β=25∘\mathrm { a } = 14 , \quad \mathrm {~b} = 12 , \quad \beta = 25 ^ { \circ }a=14, b=12,β=25∘

Two Sides and an Angle Are Given $\mathrm { a } = 14 , \quad \mathrm {~b} = 12 , \quad \beta = 25 ^ { \circ }$