(16) The angle of measure $$ \frac{31}{6} \pi $$ lies in the ......... quadrant $$ \begin{array}{llll} ext { (a) First } & ext { (b) Second } & ext { (c) Third } & ext { (d) Forth }\end{array} $$

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(16) The angle of measure $$ \frac{31}{6} \pi $$ lies in the ......... quadrant $$ \begin{array}{llll}	ext { (a) First } & 	ext { (b) Second } & 	ext { (c) Third } & 	ext { (d) Forth }\end{array} $$

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To determine the quadrant in which the angle $$ \frac{31}{6} \pi $$ lies, we first need to find its equivalent angle within the range of $$ 0 $$ to $$ 2\pi $$.

1. **Convert the angle to a more manageable form**:
   $$
   \frac{31}{6} \pi
   $$

2. **Find the equivalent angle by subtracting $$ 2\pi $$**:
   Since $$ 2\pi = \frac{12}{6} \pi $$, we can subtract $$ 2\pi $$ multiple times until the angle is within the range $$ [0, 2\pi) $$.

First, we can subtract $$ 2\pi $$ once:
   $$
   \frac{31}{6} \pi - 2\pi = \frac{31}{6} \pi - \frac{12}{6} \pi = \frac{19}{6} \pi
   $$

Next, we can subtract $$ 2\pi $$ again:
   $$
   \frac{19}{6} \pi - 2\pi = \frac{19}{6} \pi - \frac{12}{6} \pi = \frac{7}{6} \pi
   $$

3. **Determine the quadrant of $$ \frac{7}{6} \pi $$**:
   The angle $$ \frac{7}{6} \pi $$ can be expressed in degrees:
   $$
   \frac{7}{6} \pi \times \frac{180}{\pi} = 210^\circ
   $$

The angle $$ 210^\circ $$ lies in the **third quadrant** (between $$ 180^\circ $$ and $$ 270^\circ $$).

Thus, the angle $$ \frac{31}{6} \pi $$ lies in the **third quadrant**.

The answer is $$ \text{(c) Third} $$.
The angle $$ \frac{31}{6} \pi $$ lies in the third quadrant.

(16) The angle of measure \( \frac{31}{6} \pi \) lies in the ......... quadrant \( \begin{array}{llll}\text { (a) First } & \text { (b) Second } & \text { (c) Third } & \text { (d) Forth }\end{array} \)

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