We are tasked with solving a triangle using the Law of Sines. The Law of Sines states:

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

Step 1: Solving for $\sin(B)$

We are given:

• $A = 77^\circ$,
• $a = 12$,
• $b = 13$.

Using the Law of Sines: $\frac{12}{\sin(77^\circ)} = \frac{13}{\sin(B)}.$

First, calculate $\sin(77^\circ)$: $\sin(77^\circ) \approx 0.97437.$

Thus: $\frac{12}{0.97437} = \frac{13}{\sin(B)}.$

Simplify: $12.318 = \frac{13}{\sin(B)}.$

Rearrange for $\sin(B)$: $\sin(B) = \frac{13}{12.318} \approx 1.055.$

Step 2: Check for Possibility

The value of $\sin(B)$ must lie between $-1$ and $1$. Here, $\sin(B) = 1.055$, which is not possible.

Conclusion

Since $\sin(B) > 1$, a triangle with these given values is IMPOSSIBLE.

Final Answer

$B = \text{IMPOSSIBLE}, \quad C = \text{IMPOSSIBLE}, \quad c = \text{IMPOSSIBLE}.$

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Would you like further clarification or additional exploration of this problem?

Related Questions

1. What happens if $b = 11$ instead of $b = 13$?
2. How does the Law of Cosines differ in application from the Law of Sines?
3. Why is $\sin(B)$ limited to values between $-1$ and $1$?
4. Can we solve this problem graphically to confirm impossibility?
5. How does rounding affect intermediate values in trigonometric calculations?

Tip:

When solving triangles, always check if given conditions satisfy the triangle inequality theorem as a quick feasibility test!