Which of the following is the correct value of `((7pi^(c))/(4) + (4pi^(c))/(3)-(5pi^(c))/(12))`?

Options

A. `(5 pi^(c))/(4)`
B. `(5pi^(c))/(3)`
C. `300^(@)`
D. `480^(@)`

Answer

To solve the expression \(\left(\frac{7\pi}{4} + \frac{4\pi}{3} - \frac{5\pi}{12}\right)\), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
We need to find the LCM of the denominators: 4, 3, and 12.

- The LCM of 4, 3, and 12 is 12.

### Step 2: Rewrite Each Fraction with the LCM as the Denominator
Now we will convert each term to have a denominator of 12:

1. \(\frac{7\pi}{4} = \frac{7\pi \times 3}{4 \times 3} = \frac{21\pi}{12}\)
2. \(\frac{4\pi}{3} = \frac{4\pi \times 4}{3 \times 4} = \frac{16\pi}{12}\)
3. \(\frac{5\pi}{12} = \frac{5\pi}{12}\) (already has the correct denominator)

### Step 3: Combine the Fractions
Now we can combine the fractions:

\[ \frac{21\pi}{12} + \frac{16\pi}{12} - \frac{5\pi}{12} = \frac{21\pi + 16\pi - 5\pi}{12} = \frac{32\pi}{12} \]

### Step 4: Simplify the Result
Now we simplify \(\frac{32\pi}{12}\):

\[ \frac{32\pi}{12} = \frac{8\pi}{3} \quad \text{(by dividing both numerator and denominator by 4)} \]

### Step 5: Convert to Degrees
To convert \(\frac{8\pi}{3}\) radians to degrees, we use the conversion factor \(\frac{180^\circ}{\pi}\):

\[ \frac{8\pi}{3} \times \frac{180^\circ}{\pi} = \frac{8 \times 180^\circ}{3} = \frac{1440^\circ}{3} = 480^\circ \]

### Final Answer
Thus, the correct value of \(\left(\frac{7\pi}{4} + \frac{4\pi}{3} - \frac{5\pi}{12}\right)\) is \(480^\circ\).

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