In `triangleABC,angleA=30^(@),b=8, and a=4sqrt(2)`, angle C could equal

Options

A. `45^(@)`
B. `135^(@)`
C. `60^(@)`
D. `15^(@)`

Explanation

To solve the problem, we will use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Given:
- Angle A = 30°
- Side a = 4√2
- Side b = 8

We need to find the possible values for angle C.

### Step 1: Use the Law of Sines

According to the Law of Sines:

\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \]

We can focus on the first two parts of the equation:

\[ \frac{\sin A}{a} = \frac{\sin B}{b} \]

Substituting the known values:

\[ \frac{\sin 30°}{4\sqrt{2}} = \frac{\sin B}{8} \]

### Step 2: Substitute the value of sin 30°

We know that:

\[ \sin 30° = \frac{1}{2} \]

So we can substitute this into the equation:

\[ \frac{\frac{1}{2}}{4\sqrt{2}} = \frac{\sin B}{8} \]

### Step 3: Simplify the left side

Calculating the left side:

\[ \frac{1}{2} \div 4\sqrt{2} = \frac{1}{8\sqrt{2}} \]

Thus, we have:

\[ \frac{1}{8\sqrt{2}} = \frac{\sin B}{8} \]

### Step 4: Cross-multiply to solve for sin B

Cross-multiplying gives:

\[ \sin B = \frac{1}{8\sqrt{2}} \cdot 8 = \frac{1}{\sqrt{2}} \]

### Step 5: Determine the possible angles for B

The sine function can yield two angles for \(\sin B = \frac{1}{\sqrt{2}}\):

1. \(B = 45°\)
2. \(B = 135°\)

### Step 6: Calculate angle C for each case

Using the triangle angle sum property, we have:

\[ A + B + C = 180° \]

#### Case 1: If \(B = 45°\)

\[ 30° + 45° + C = 180° \]
\[ C = 180° - 75° = 105° \]

#### Case 2: If \(B = 135°\)

\[ 30° + 135° + C = 180° \]
\[ C = 180° - 165° = 15° \]

### Conclusion

The possible values for angle C are:

- \(C = 105°\)
- \(C = 15°\)