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In the triangle ABC, a = 5, A = 30° and B = 80°. Calculate C, b and c.
do it like that example

To solve the triangle problem where \( a = 5 \), \( A = 30^\circ \), and \( B = 80^\circ \), let's proceed step by step as in the provided example.

### Step 1: Calculate the value of angle \( C \)
Since the sum of the angles in any triangle is \( 180^\circ \), we can find \( C \) as:
\[
C = 180^\circ - A - B = 180^\circ - 30^\circ - 80^\circ = 70^\circ
\]

### Step 2: Use the sine rule to calculate side \( b \)
The sine rule states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
We use the part involving sides \( a \) and \( b \):
\[
\frac{b}{\sin B} = \frac{a}{\sin A}
\]
Substituting the known values:
\[
\frac{b}{\sin 80^\circ} = \frac{5}{\sin 30^\circ}
\]

### Step 3: Transpose the equation to solve for \( b \)
\[
b = \frac{5 \times \sin 80^\circ}{\sin 30^\circ}
\]

### Step 4: Evaluate \( b \)
Calculate the sine values and compute \( b \):
\[
\sin 80^\circ \approx 0.9848,\quad \sin 30^\circ = 0.5
\]
\[
b = \frac{5 \times 0.9848}{0.5} \approx \frac{4.924}{0.5} = 9.848
\]
So, \( b \approx 9.85 \) (rounded to two decimal places).

### Step 5: Use the sine rule to calculate side \( c \)
Again using the sine rule:
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]
Substituting the known values:
\[
\frac{c}{\sin 70^\circ} = \frac{5}{\sin 30^\circ}
\]

### Step 6: Transpose the equation to solve for \( c \)
\[
c = \frac{5 \times \sin 70^\circ}{\sin 30^\circ}
\]

### Step 7: Evaluate \( c \)
Calculate the sine values and compute \( c \):
\[
\sin 70^\circ \approx 0.9397
\]
\[
c = \frac{5 \times 0.9397}{0.5} = \frac{4.6985}{0.5} = 9.397
\]
So, \( c \approx 9.40 \) (rounded to two decimal places).

### Summary of Results:
- \( C = 70^\circ \)
- \( b \approx 9.85 \)
- \( c \approx 9.40 \)

Would you like to dive deeper into any of these steps or have any additional questions?

### Follow-up Questions:
1. How does the sine rule apply to non-right-angled triangles?
2. Can the sine rule be used to find the area of a triangle?
3. How would the process change if two sides and the included angle were given?
4. What happens if the given triangle is obtuse-angled?
5. Can the cosine rule be used in place of the sine rule for this problem?

### Tip:
When solving for sides using the sine rule, always double-check the angle opposite to the side you are calculating to ensure accuracy.

Learn how to calculate the angles and sides of triangle ABC where side a = 5 units, angle A = 30°, and angle B = 80°. Step-by-step solution using trigonometric principles and the sine rule.

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