In `DeltaABC , angleA=(2pi^(c))/(3)` and `angleB=45^(@)`. Find ` angleC` in both the systems.

Answer

To find angle C in triangle ABC where angle A = \( \frac{2\pi}{3} \) radians and angle B = \( 45^\circ \), we can follow these steps:

### Step 1: Convert angle A from radians to degrees
We know that \( \pi \) radians is equal to \( 180^\circ \). Therefore, we can convert angle A as follows:

\[ \text{Angle A} = \frac{2\pi}{3} \text{ radians} = \frac{2 \times 180^\circ}{3} = 120^\circ \]

### Step 2: Use the triangle angle sum property
In any triangle, the sum of the angles is \( 180^\circ \). Therefore, we can write:

\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]

Substituting the known values:

\[ 120^\circ + 45^\circ + \text{Angle C} = 180^\circ \]

### Step 3: Solve for angle C
Now, we can simplify the equation:

\[ 165^\circ + \text{Angle C} = 180^\circ \]

Subtracting \( 165^\circ \) from both sides gives:

\[ \text{Angle C} = 180^\circ - 165^\circ = 15^\circ \]

### Step 4: Convert angle C from degrees to radians
To convert \( 15^\circ \) to radians, we use the conversion factor \( \frac{\pi \text{ radians}}{180^\circ} \):

\[ \text{Angle C} = 15^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{15\pi}{180} = \frac{\pi}{12} \text{ radians} \]

### Final Result
Thus, the angles in both systems are:
- Angle C in degrees: \( 15^\circ \)
- Angle C in radians: \( \frac{\pi}{12} \)

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