To solve for side $a$ of triangle $ABC$, we will use the Law of Sines, which states:

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

Step 1: Write down given values

• $A = 80^\circ$
• $B = 30^\circ$
• $b = 13 \, \text{cm}$

Since the angles of a triangle sum to $180^\circ$, we can find $C$ as: $C = 180^\circ - A - B = 180^\circ - 80^\circ - 30^\circ = 70^\circ$

Step 2: Apply the Law of Sines

Using the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$ Substitute the known values: $\frac{a}{\sin 80^\circ} = \frac{13}{\sin 30^\circ}$

Step 3: Solve for $a$

Rearranging for $a$: $a = \frac{\sin 80^\circ}{\sin 30^\circ} \cdot 13$

• $\sin 80^\circ \approx 0.9848$
• $\sin 30^\circ = 0.5$

Substitute these values: $a = \frac{0.9848}{0.5} \cdot 13$ $a \approx 25.605$

Thus, $a \approx 25.6 \, \text{cm}$.

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Final Answer:

The length of side $a$ is approximately 25.6 cm.

Would you like additional details or have any questions?

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Related Questions:

1. How would the solution change if angle $A$ were larger than $90^\circ$?
2. How can you verify this calculation using the cosine rule instead?
3. Why is the sine of $30^\circ$ always $0.5$, regardless of the triangle?
4. Can the Law of Sines solve for angles as well as sides?
5. How does rounding trigonometric values affect precision in solutions?

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Tip:

When using the Law of Sines, always ensure that the triangle is not ambiguous (e.g., for SSA cases, there may be two solutions).