Which of the following is equivalent to the value of $(\frac{5\pi}{9}c + \frac{13\pi}{18}c - \frac{17\pi}{6}c)$ when converted to degrees? $\newline$ (A) $20^\circ$ $\newline$ (B) $80^\circ$ $\newline$ (C) $280^\circ$ $\newline$ (D) $320^\circ$

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Algebra 2

Convert between radians and degrees

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Q. Which of the following is equivalent to the value of

(\frac{5\pi}{9}c + \frac{13\pi}{18}c - \frac{17\pi}{6}c)

when converted to degrees?

\newline

(A)

20^\circ

\newline

(B)

80^\circ

\newline

(C)

280^\circ

\newline

(D)

320^\circ

Convert to degrees: Convert each term in the expression from radians to degrees. $\newline$ To convert radians to degrees, we use the formula: $\text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right)$ . $\newline$ Let's convert each term separately. $\newline$ For the first term, $\frac{5\pi}{9c}$ , we have: $\left(\frac{5\pi}{9c}\right) \times \left(\frac{180}{\pi}\right) = \left(\frac{5 \times 180}{9}\right)c = \left(5 \times 20\right)c = 100c$ degrees.
Convert second term: Convert the second term, $\frac{13\pi}{18}c$ , to degrees. $\newline$ Using the same conversion formula: $\left(\frac{13\pi}{18}c\right) \times \left(\frac{180}{\pi}\right) = \left(\frac{13 \times 180}{18}\right)c = (13 \times 10)c = 130c$ degrees.
Convert third term: Convert the third term, $\frac{17\pi}{6}c$ , to degrees. $\newline$ Again, using the conversion formula: $\left(\frac{17\pi}{6}c\right) \times \left(\frac{180}{\pi}\right) = \left(\frac{17 \times 180}{6}\right)c = (17 \times 30)c = 510c$ degrees.
Combine converted terms: Combine the converted terms to find the total value in degrees. $\newline$ Now we add the first two terms and subtract the third term: $100c$ degrees + $130c$ degrees - $510c$ degrees. $\newline$ This simplifies to: $(100 + 130 - 510)c$ degrees = $(230 - 510)c$ degrees = $-280c$ degrees. $\newline$ Since $c$ is a common factor, it cancels out, leaving us with $-280$ degrees.
Determine matching option: Determine which of the given options matches the calculated degree value. $\newline$ The calculated value is $-280$ degrees. However, the options are all positive degrees. We know that adding or subtracting full rotations ( $360$ degrees) does not change the angle's position. Therefore, we can add $360$ degrees to $-280$ degrees to find an equivalent positive angle. $\newline$ $-280$ degrees + $360$ degrees = $80$ degrees.
Choose correct option: Choose the correct option that matches the calculated degree value. $\newline$ The correct option that matches $80$ degrees is option B.