Question

$\qquad 4\left[\frac{82\pi}{7}-\frac{18\pi}{7}\right]\left(\cos\frac{3\pi}{7}\right)=?$

Answer 1

$\frac{256\pi}{7}\cos\frac{3\pi}{7}$

Answer 2

To solve the given expression: $4\left[\frac{82\pi}{7}-\frac{18\pi}{7}\right]\left(\cos\frac{3\pi}{7}\right)$

First, simplify the terms inside the square brackets: $\frac{82\pi}{7}-\frac{18\pi}{7} = \frac{(82-18)\pi}{7} = \frac{64\pi}{7}$

Now, substitute this simplified term back into the expression: $4\left[\frac{64\pi}{7}\right]\left(\cos\frac{3\pi}{7}\right)$

Multiply the numerical coefficients: $4 \times \frac{64\pi}{7} = \frac{256\pi}{7}$

Finally, multiply by the trigonometric term: $\frac{256\pi}{7}\cos\frac{3\pi}{7}$

The value of $\cos\frac{3\pi}{7}$ is an irrational number and does not simplify further in a way that would cancel out $\pi$ or the denominator 7 to yield a simple rational number. Therefore, the expression is simplified to its most compact form.

The final answer is $\frac{256\pi}{7}\cos\frac{3\pi}{7}$.

Explanation of the solution:

1. Simplify the bracketed term: Combine the fractions with a common denominator: $\frac{82\pi}{7} - \frac{18\pi}{7} = \frac{(82-18)\pi}{7} = \frac{64\pi}{7}$.
2. Multiply by the leading constant: Multiply the result from step 1 by 4: $4 \times \frac{64\pi}{7} = \frac{256\pi}{7}$.
3. Combine with the trigonometric term: Multiply the result from step 2 by $\cos\frac{3\pi}{7}$ to get the final simplified expression: $\frac{256\pi}{7}\cos\frac{3\pi}{7}$.