The image depicts a right triangle $\triangle ABC$ with the following measurements:

• $\angle C = 90^\circ$ (a right angle).
• $\overline{AC} = 5 \, \text{cm}$.
• $\overline{AB} = 10 \, \text{cm}$.
• $\angle A = 30^\circ$.
• $\overline{BC} = b$ (the unknown side).

To solve for $b$ (side $BC$), we can use trigonometry, specifically the sine or cosine functions, since angle $A = 30^\circ$ and the sides adjacent or opposite to that angle are given.

1. Using the sine function: The sine of an angle in a right triangle is defined as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ For $\angle A = 30^\circ$, the side opposite is $BC = b$, and the hypotenuse is $AB = 10 \, \text{cm}$. $\sin(30^\circ) = \frac{b}{10}$ Since $\sin(30^\circ) = 0.5$, we get: $0.5 = \frac{b}{10}$ Solving for $b$: $b = 10 \times 0.5 = 5 \, \text{cm}$

Thus, the correct equation to solve for $b$ is: $\sin(30^\circ) = \frac{b}{10}$

Would you like more details on trigonometry or help with other related concepts?

Here are 5 related questions to consider:

1. What is the cosine equation for this triangle?
2. How can we solve for the third side using the Pythagorean theorem?
3. What is the angle $B$ in triangle $\triangle ABC$?
4. How can trigonometry be used to solve other right triangles?
5. What are some real-world applications of trigonometry in geometry?

Tip: In right triangles, always identify the given angle and its relationship to the known sides (opposite, adjacent, or hypotenuse) to choose the correct trigonometric function.