In the triangle ABC, a = 5, A = 30° and B = 80°. Calculate C, b and c.

To solve the triangle \(ABC\) given \(a = 5\), \(\angle A = 30^\circ\), and \(\angle B = 80^\circ\), you need to determine the remaining angle \(\angle C\) and the sides \(b\) and \(c\). Here's the step-by-step process:

### 1. **Calculate \(\angle C\)**:
Since the sum of the angles in a triangle is \(180^\circ\), you can find \(\angle C\) as follows:
\[
\angle C = 180^\circ - \angle A - \angle B
\]
\[
\angle C = 180^\circ - 30^\circ - 80^\circ = 70^\circ
\]

### 2. **Use the Law of Sines**:
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

### 3. **Calculate side \(b\)**:
Use the Law of Sines to find \(b\):
\[
\frac{b}{\sin B} = \frac{a}{\sin A}
\]
\[
b = \frac{a \cdot \sin B}{\sin A}
\]
Substitute the given values:
\[
b = \frac{5 \cdot \sin(80^\circ)}{\sin(30^\circ)}
\]
\[
\sin(80^\circ) \approx 0.9848,\quad \sin(30^\circ) = 0.5
\]
\[
b = \frac{5 \cdot 0.9848}{0.5} = 9.848
\]

### 4. **Calculate side \(c\)**:
Now, use the Law of Sines to find \(c\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]
\[
c = \frac{a \cdot \sin C}{\sin A}
\]
\[
c = \frac{5 \cdot \sin(70^\circ)}{\sin(30^\circ)}
\]
\[
\sin(70^\circ) \approx 0.9397
\]
\[
c = \frac{5 \cdot 0.9397}{0.5} = 9.397
\]

### Final Answers:
- \(\angle C = 70^\circ\)
- \(b \approx 9.848\)
- \(c \approx 9.397\)

Would you like a more detailed explanation of any of these steps?

Here are 5 follow-up questions you might consider:

1. How would you solve this triangle if you were only given sides \(a\) and \(b\) and angle \(A\)?
2. What would happen if \(\angle A\) and \(\angle B\) were very close to 90 degrees?
3. How does the Law of Cosines compare to the Law of Sines for solving triangles?
4. What would change if this triangle were non-Euclidean, such as on a spherical surface?
5. How do you verify your solution using trigonometric identities?

**Tip:** Always ensure your calculator is set to the correct mode (degrees/radians) when performing trigonometric calculations.

Learn how to solve triangle ABC with given sides and angles: a = 5, A = 30°, B = 80°. Discover the value of angle C, side b, and side c using the Law of Sines and trigonometric calculations. Suitable for high school students.

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