If `a=2b` and `A=3B`, find the angles A, B, C of `DeltaABC` :

Options

A. `90^(@), 30^(@), 60^(@)`
B. `30^(@), 60^(@), 90^(@)`
C. `45^(@), 45^(@), 90^(@)`
D. data insufficient

Answer

To solve the problem, we need to find the angles A, B, and C of triangle ABC given the relationships between the sides and angles: \( a = 2b \) and \( A = 3B \).

### Step-by-Step Solution:

1. **Identify the Relationships**:
We are given:
- \( a = 2b \) (where \( a \) and \( b \) are the lengths of the sides opposite angles A and B respectively)
- \( A = 3B \) (where \( A \) and \( B \) are the angles of the triangle)

2. **Use the Law of Sines**:
According to the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Substituting the values of \( a \) and \( A \):
\[ \frac{2b}{\sin(3B)} = \frac{b}{\sin B} \]

3. **Simplify the Equation**:
We can cancel \( b \) from both sides (assuming \( b \neq 0 \)):
\[ \frac{2}{\sin(3B)} = \frac{1}{\sin B} \]
This leads to:
\[ 2 \sin B = \sin(3B) \]

4. **Use the Sine Triple Angle Formula**:
The sine triple angle formula states:
\[ \sin(3B) = 3\sin B - 4\sin^3 B \]
Substituting this into our equation gives:
\[ 2 \sin B = 3 \sin B - 4 \sin^3 B \]

5. **Rearranging the Equation**:
Rearranging the equation:
\[ 0 = 3 \sin B - 4 \sin^3 B - 2 \sin B \]
This simplifies to:
\[ 0 = \sin B (1 - 4 \sin^2 B) \]

6. **Finding Possible Values for \( \sin B \)**:
From the equation, we have two cases:
- \( \sin B = 0 \)
- \( 1 - 4 \sin^2 B = 0 \)

For \( \sin B = 0 \), \( B \) cannot be 0 degrees as it would not form a triangle.

For \( 1 - 4 \sin^2 B = 0 \):
\[ 4 \sin^2 B = 1 \implies \sin^2 B = \frac{1}{4} \implies \sin B = \frac{1}{2} \]
Therefore, \( B = 30^\circ \).

7. **Finding Angle A**:
Since \( A = 3B \):
\[ A = 3 \times 30^\circ = 90^\circ \]

8. **Finding Angle C**:
The sum of angles in a triangle is \( 180^\circ \):
\[ A + B + C = 180^\circ \implies 90^\circ + 30^\circ + C = 180^\circ \]
Thus:
\[ C = 180^\circ - 120^\circ = 60^\circ \]

### Final Angles:
- \( A = 90^\circ \)
- \( B = 30^\circ \)
- \( C = 60^\circ \)

### Summary:
The angles of triangle ABC are:
- \( A = 90^\circ \)
- \( B = 30^\circ \)
- \( C = 60^\circ \)