$Identifying Quadrants on the Unit Circle$

$Identify\ the\ quadrant\ in\ which\ the\ angle\ \theta = 5\pi/4\ \text{radians}\ lies\ on\ the\ unit\ circle.$

$To\ determine\ the\ quadrant\ of\ \theta = 5\pi/4:\ $

$1. \text{Convert\ the\ angle\ to\ degrees\ for\ better\ understanding:}$

$\theta = \frac{5\cdot180}{4} = 225^{\circ}\ $

$2. \text{Analyze\ the\ degree\ measure:}$

$0^{\circ} \leq 225^{\circ} \leq 360^{\circ}\ $

$225^{\circ} \text{lies\ between\ 180^{\circ}\ (negative\ x-axis)\ and\ 270^{\circ}\ (negative\ y-axis),\ which\ is\ the\ Third\ Quadrant.}$

$Therefore,\ \theta = 5\pi/4\ \text{radians\ lies\ in\ the\ Third\ Quadrant.}$

$Find the quadrant for the angle heta = -pi/6 ext{radians} on the unit circle.$

$To determine the quadrant of heta = -pi/6:$

$1. ext{Convert the angle to degrees: }$

$ heta = -frac{180}{6} = -30^{circ} $

$2. ext{Since the angle is negative, move clockwise from the positive x-axis:}$

$-30^{circ} ext{is 30 degrees clockwise from the positive x-axis, which places it in the Fourth Quadrant.}$

$Therefore, heta = -pi/6 ext{radians lies in the Fourth Quadrant.}$

$Identify the quadrant of the angle heta = 7pi/6 ext{radians} on the unit circle.$

$1. ext{Convert to degrees:} $

$ heta = 7pi/6 cdot frac{180}{pi} = 210^{circ} $

$2. ext{Since 210^{circ} is between 180^{circ} and 270^{circ}, it lies in the Third Quadrant.}$

$Therefore, heta = 7pi/6 ext{is in the Third Quadrant.}$

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