Quadrants for Angles α and β: QI and QII

Answer

The angle of rotation $\alpha$ has its terminal ray in Quadrant I (QI), while the angle of rotation $\beta$ has its terminal ray in Quadrant II (QII).

Explanation

Based on the provided image, we observe a circle centered at the origin $C(0,0)$ with a radius of $10$. Two terminal rays are shown: ray $CB$ passing through point $B(6,8)$ and ray $CB'$ passing through point $B'(-6,8)$.

1. Analyze the Cartesian Coordinate System The coordinate plane is divided into four quadrants based on the signs of the $x$ and $y$ coordinates:

   Quadrant	x-coordinate	y-coordinate	Angle Range
   QI	Positive (+)	Positive (+)	$0^\circ < \theta < 90^\circ$
   QII	Negative (-)	Positive (+)	$90^\circ < \theta < 180^\circ$
   QIII	Negative (-)	Negative (-)	$180^\circ < \theta < 270^\circ$
   QIV	Positive (+)	Negative (-)	$270^\circ < \theta < 360^\circ$
   The table defines the standard regions of a 2D plane used for trigonometric rotations.

2. Locate angle $\alpha$ Angle $\alpha$ is the rotation from the positive x-axis to the terminal ray $CB$. Point $B$ is given as $(6, 8)$. Since both the $x$-coordinate ($6$) and the $y$-coordinate ($8$) are positive, the ray lies in the upper-right section of the graph. ⚠️ This step is required on exams: Always verify the signs of the coordinates to determine the quadrant. $\text{Point } B(x, y) = (6, 8) \rightarrow x > 0, y > 0$ This indicates that the point is located in the first quadrant.

3. Locate angle $\beta$ Angle $\beta$ is the rotation from the positive x-axis to the terminal ray $CB'$. By observing the symmetry in the diagram, point $B'$ has coordinates $(-6, 8)$. In this case, the $x$-coordinate is negative while the $y$-coordinate remains positive. $\text{Point } B'(x, y) = (-6, 8) \rightarrow x < 0, y > 0$ The combination of a negative $x$ and a positive $y$ places the terminal side in the upper-left section.

Final Answer

$\text{Angle } \alpha: \boxed{\text{QI}}$ $\text{Angle } \beta: \boxed{\text{QII}}$

Common Mistakes

• Confusing Angle Measure with Quadrant: Students sometimes think that because $\beta$ is a large angle, it must be in QIII or QIV. Always look at the final position of the ray relative to the axes.
• Incorrect Origin Reference: Ensure you are measuring the rotation starting from the positive x-axis (Standard Position). Measuring from the y-axis will lead to incorrect quadrant identification.