In the triangle ABC, c = 27, C = 42° and A = 105°. Calculate B, a and b.

To solve the triangle problem where \( c = 27 \), \( C = 42^\circ \), and \( A = 105^\circ \), we'll follow a step-by-step approach similar to the previous example.

### Step 1: Calculate the value of angle \( B \)
Since the sum of the angles in any triangle is \( 180^\circ \), we can find \( B \) as:
\[
B = 180^\circ - A - C = 180^\circ - 105^\circ - 42^\circ = 33^\circ
\]

### Step 2: Use the sine rule to calculate side \( a \)
The sine rule states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
We use the part involving sides \( a \) and \( c \):
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the known values:
\[
\frac{a}{\sin 105^\circ} = \frac{27}{\sin 42^\circ}
\]

### Step 3: Transpose the equation to solve for \( a \)
\[
a = \frac{27 \times \sin 105^\circ}{\sin 42^\circ}
\]

### Step 4: Evaluate \( a \)
Calculate the sine values and compute \( a \):
\[
\sin 105^\circ \approx 0.9659, \quad \sin 42^\circ \approx 0.6691
\]
\[
a = \frac{27 \times 0.9659}{0.6691} \approx \frac{26.0593}{0.6691} \approx 38.94
\]
So, \( a \approx 38.94 \) (rounded to two decimal places).

### Step 5: Use the sine rule to calculate side \( b \)
Again using the sine rule:
\[
\frac{b}{\sin B} = \frac{c}{\sin C}
\]
Substituting the known values:
\[
\frac{b}{\sin 33^\circ} = \frac{27}{\sin 42^\circ}
\]

### Step 6: Transpose the equation to solve for \( b \)
\[
b = \frac{27 \times \sin 33^\circ}{\sin 42^\circ}
\]

### Step 7: Evaluate \( b \)
Calculate the sine values and compute \( b \):
\[
\sin 33^\circ \approx 0.5446
\]
\[
b = \frac{27 \times 0.5446}{0.6691} \approx \frac{14.7042}{0.6691} \approx 21.98
\]
So, \( b \approx 21.98 \) (rounded to two decimal places).

### Summary of Results:
- \( B = 33^\circ \)
- \( a \approx 38.94 \)
- \( b \approx 21.98 \)

Would you like further explanation on any of these steps or have any additional questions?

### Follow-up Questions:
1. How does knowing the largest angle affect the choice of the sine rule?
2. What would be the next step if the problem asked for the area of triangle ABC?
3. How would you solve the triangle if two sides and the included angle were given instead?
4. What is the significance of the sine rule when the triangle is acute-angled?
5. Can you verify the results using the cosine rule?

### Tip:
Always ensure that the sum of the angles adds up to \( 180^\circ \) as a way to verify that the calculated angles are correct.

Learn how to calculate angles and sides of triangle ABC with given sides and angles using the sine rule. Solve for angle B, sides a and b with specific values: c = 27, C = 42°, A = 105°. Step-by-step solution included.

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