To solve the given triangle problem using the Sine Law, let's analyze the image.

Problem Analysis:

You have a triangle $\triangle ABC$ with the following information provided:

• $\angle B = 30^\circ$
• Side $AB = 8$ units
• Side $AC = 10$ units

You need to determine the value of $\angle A$ using the Sine Law.

Sine Law Formula:

The Sine Law states:

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

Where:

• $a$, $b$, and $c$ are the sides of the triangle.
• $A$, $B$, and $C$ are the angles opposite to sides $a$, $b$, and $c$, respectively.

For this triangle:

• $a = 10$ (opposite $\angle B$)
• $b = 8$ (opposite $\angle A$)
• $\angle B = 30^\circ$

Step-by-Step Solution:

Using the Sine Law, we have:

$\frac{10}{\sin 30^\circ} = \frac{8}{\sin A}$

Substituting $\sin 30^\circ = 0.5$:

$\frac{10}{0.5} = \frac{8}{\sin A}$

This simplifies to:

$20 = \frac{8}{\sin A}$

Now, solving for $\sin A$:

$\sin A = \frac{8}{20} = 0.4$

Finally, $\angle A$ can be determined using the inverse sine function:

$A = \sin^{-1}(0.4)$

Let's compute the exact value.

Calculation:

I will now calculate the angle.The value of $\angle A$ is approximately $23.58^\circ$.

Would you like more details on the solution or have any other questions?

Here are 5 related questions that could extend this topic:

1. How can you apply the Cosine Law to check the accuracy of the result?
2. How do you determine the remaining angle $\angle C$ in this triangle?
3. What is the importance of using the Sine Law in non-right triangles?
4. How does the Sine Law relate to vector quantities in physics?
5. What are the conditions under which the Sine Law cannot be applied?

Tip: Always double-check your calculations by using alternative methods (like the Cosine Law) when possible, especially in trigonometry problems.