coterminal angle, quadrant, and reference angle for a given negative rotation

1. Find the coterminal angle: To find a coterminal angle within the range $0^\circ \leq \theta < 360^\circ$, we add multiples of $360^\circ$ to the given angle $-969^\circ$ until we get an angle in the desired range.

We can add $360^\circ$ repeatedly:
$-969^\circ + 360^\circ = -609^\circ$
$-609^\circ + 360^\circ = -249^\circ$
$-249^\circ + 360^\circ = 111^\circ$

Alternatively, we can divide $-969$ by $360$ to see how many full rotations are involved:
$$\frac{-969}{360} \approx -2.69$$
This means we need to add $3$ full rotations (since we are dealing with a negative angle, we round up to the next whole number of rotations to get a positive angle).
$$ -969^\circ + 3 \times 360^\circ = -969^\circ + 1080^\circ = 111^\circ $$
The coterminal angle is $111^\circ$.

1. Determine the quadrant: The angle $111^\circ$ is greater than $90^\circ$ and less than $180^\circ$. Therefore, it lies in Quadrant II.

3. Find the reference angle: The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. For an angle in Quadrant II, the reference angle is calculated as $180^\circ - \theta$.
$$ \text{Reference Angle} = 180^\circ - 111^\circ = 69^\circ $$