Question

1. （triangle with side 15, side 31, angle 126°, and angle x°）

Explanation:

To solve for \( x \) in the triangle, we can use the Law of Sines, which states that for a triangle with sides \( a \), \( b \), \( c \) opposite angles \( A \), \( B \), \( C \) respectively:

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

Step 1: Identify sides and angles

Let’s label the triangle:

• Side \( a = 15 \) (opposite angle \( A = x^\circ \))
• Side \( b = 31 \) (opposite angle \( B = 126^\circ \))

Step 2: Apply the Law of Sines

Using the Law of Sines:

\[
\frac{15}{\sin x} = \frac{31}{\sin 126^\circ}
\]

Solve for \( \sin x \):

\[
\sin x = \frac{15 \cdot \sin 126^\circ}{31}
\]

Step 3: Calculate \( \sin 126^\circ \)

\( \sin 126^\circ = \sin(180^\circ - 54^\circ) = \sin 54^\circ \approx 0.8090 \)

Step 4: Substitute and solve for \( \sin x \)

\[
\sin x = \frac{15 \cdot 0.8090}{31} \approx \frac{12.135}{31} \approx 0.3915
\]

Step 5: Find \( x \)

Take the inverse sine:

\[
x = \arcsin(0.3915) \approx 22.9^\circ \approx 23^\circ
\]

Answer:

\( x \approx \boldsymbol{23^\circ} \) (or \( 22.9^\circ \) for more precision)