To determine whether there are zero, one, or two triangles possible with the given values [tex]\( m\angle A = 48^\circ \)[/tex], [tex]\( a = 10 \)[/tex] m, and [tex]\( b = 12 \)[/tex] m, we follow these steps:
1. Determine the relationship of the sides and the sine of angle [tex]\( A \)[/tex]:
Using the Law of Sines:
[tex]\[
\frac{\sin(A)}{a} = \frac{\sin(B)}{b}
\][/tex]
Plug in the given values:
[tex]\[
\frac{\sin(48^\circ)}{10} = \frac{\sin(B)}{12}
\][/tex]
2. Calculate [tex]\(\sin(48^\circ)\)[/tex]:
[tex]\[
\sin(48^\circ) \approx 0.7431
\][/tex]
So, the equation becomes:
[tex]\[
\frac{0.7431}{10} = \frac{\sin(B)}{12}
\][/tex]
3. Solve for [tex]\(\sin(B)\)[/tex]:
[tex]\[
\sin(B) = 12 \times \frac{0.7431}{10}
= 12 \times 0.07431
= 0.89172
\][/tex]
4. Check if [tex]\(\sin(B)\)[/tex] is within the valid range for sine values (0 to 1):
Since [tex]\( \sin(B) = 0.89172 \)[/tex] which is within [tex]\([0, 1]\)[/tex], we can find angle [tex]\( B \)[/tex].
5. Find angle [tex]\( B \)[/tex]:
Using the arcsine function:
[tex]\[
B = \arcsin(0.89172) \approx 63.1^\circ
\][/tex]
6. Calculate the third angle [tex]\( C \)[/tex]:
Use the fact that the sum of angles in a triangle is [tex]\( 180^\circ \)[/tex]:
[tex]\[
C = 180^\circ - A - B
= 180^\circ - 48^\circ - 63.1^\circ
= 68.9^\circ
\][/tex]
7. Consider the possibility of a second triangle:
In the case of the ambiguous case (SSA condition), angle [tex]\( B \)[/tex] could potentially have a second solution since [tex]\(\sin(B) = \sin(180^\circ - B)\)[/tex]:
[tex]\[
B_2 = 180^\circ - 63.1^\circ
= 116.9^\circ
\][/tex]
Check the corresponding third angle [tex]\( C_2 \)[/tex]:
[tex]\[
C_2 = 180^\circ - A - B_2
= 180^\circ - 48^\circ - 116.9^\circ
= 15.1^\circ
\][/tex]
8. Validate the second set of angles:
Both [tex]\( B_2 = 116.9^\circ \)[/tex] and [tex]\( C_2 = 15.1^\circ \)[/tex] are valid angles, meaning the second set of angles does form a triangle.
Hence, there are two possible triangles that can be formed with the given values.
