Question

Determine if there are zero, one, or two triangles for the following:
lower m angle A equals 48 degrees
a = 10 m
b = 12 m

Answer 1

To determine the number of triangles that can be formed with the given information, we can use the Law of Sines and the properties of triangles.

Given:
- Angle \( A = 48^\circ \)
- Side \( a = 10 \, m \) (the side opposite angle \( A \))
- Side \( b = 12 \, m \) (the side opposite angle \( B \))

We apply the Law of Sines:

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Substituting the known values:

\[
\frac{10}{\sin 48^\circ} = \frac{12}{\sin B}
\]

Now we can isolate \( \sin B \):

\[
\sin B = \frac{12 \cdot \sin 48^\circ}{10}
\]

Calculating \( \sin 48^\circ \):

\[
\sin 48^\circ \approx 0.7431
\]

Now, substituting this value back into our equation:

\[
\sin B \approx \frac{12 \cdot 0.7431}{10} \approx \frac{8.9172}{10} \approx 0.89172
\]

Next, we check if \( \sin B \) is valid. The range for sine values is between \( -1 \) and \( 1 \), so \( \sin B \approx 0.89172 \) is a viable value.

Now, we can find angle \( B \):

\[
B = \sin^{-1}(0.89172)
\]

Calculating \( B \):

\[
B \approx 63.6^\circ
\]

Since \( B \) is a valid angle, we should also consider the possibility of a second triangle. The second triangle occurs when we have:

\[
B' = 180^\circ - B \approx 180^\circ - 63.6^\circ = 116.4^\circ
\]

Next, we check if angle \( C \) for the second triangle is valid by ensuring the angles sum up to less than \( 180^\circ \):

For the first triangle:

\[
C = 180^\circ - A - B \approx 180^\circ - 48^\circ - 63.6^\circ \approx 68.4^\circ
\]

This angle is valid.

For the second triangle:

\[
C' = 180^\circ - A - B' \approx 180^\circ - 48^\circ - 116.4^\circ \approx 15.6^\circ
\]

This angle is also valid.

Since both triangles are valid:

**Conclusion: There are two triangles for the given information.**