In a triangle ABC, c = 2, A = `45^(@)`, a = `2 sqrt(2)`, than what is C equal to ?

Options

A. `30^(@)`
B. `15^(@)`
C. `45^(@)`
D. None of these

Explanation

To find the angle \( C \) in triangle \( ABC \) given \( c = 2 \), \( A = 45^\circ \), and \( a = 2\sqrt{2} \), we can use the Law of Sines, which states:

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

### Step 1: Write down the known values
We have:
- \( a = 2\sqrt{2} \)
- \( A = 45^\circ \)
- \( c = 2 \)

### Step 2: Substitute the known values into the Law of Sines
Using the Law of Sines, we substitute the values we know:

\[ \frac{2\sqrt{2}}{\sin 45^\circ} = \frac{2}{\sin C} \]

### Step 3: Calculate \( \sin 45^\circ \)
We know that:

\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 4: Substitute \( \sin 45^\circ \) into the equation
Now substituting \( \sin 45^\circ \) into the equation, we have:

\[ \frac{2\sqrt{2}}{\frac{1}{\sqrt{2}}} = \frac{2}{\sin C} \]

### Step 5: Simplify the left side
The left side simplifies as follows:

\[ 2\sqrt{2} \cdot \sqrt{2} = 2 \cdot 2 = 4 \]

So, we have:

\[ 4 = \frac{2}{\sin C} \]

### Step 6: Cross-multiply to solve for \( \sin C \)
Cross-multiplying gives:

\[ 4 \sin C = 2 \]

### Step 7: Solve for \( \sin C \)
Dividing both sides by 4:

\[ \sin C = \frac{2}{4} = \frac{1}{2} \]

### Step 8: Find angle \( C \)
The angle whose sine is \( \frac{1}{2} \) is:

\[ C = 30^\circ \]

### Final Answer
Thus, the value of angle \( C \) is \( 30^\circ \).

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