Question

find the measures of two angles, one positive and one negative, that are coterminal with the given angle.
\\( \frac { 7 \pi } { 9 } \\)
\\( \frac { 11 \pi } { 9 } ; - \frac { 25 \pi } { 9 } \\)
\\( \frac { 16 \pi } { 9 } ; - \frac { 11 \pi } { 9 } \\)
\\( \frac { 16 \pi } { 9 } ; - \frac { 16 \pi } { 9 } \\)
\\( \frac { 25 \pi } { 9 } ; - \frac { 11 \pi } { 9 } \\)

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles are given by \(\theta + 2k\pi\), where \(k\in\mathbb{Z}\). For \(\theta=\frac{7\pi}{9}\), when \(k = 1\):

$$ \frac{7\pi}{9}+2\pi=\frac{7\pi}{9}+\frac{18\pi}{9}=\frac{7 + 18}{9}\pi=\frac{25\pi}{9} $$

Step2: When \(k=- 1\)

$$ \frac{7\pi}{9}-2\pi=\frac{7\pi}{9}-\frac{18\pi}{9}=\frac{7-18}{9}\pi=-\frac{11\pi}{9} $$

Answer:

\(\frac{25\pi}{9};-\frac{11\pi}{9}\) (corresponding to the fourth option)