Questions: Name the quadrant and reference angle (in radians) for the angle 26π/7. Quadrant: Reference angle:

Solution Steps

To determine the quadrant and reference angle for the given angle \(\frac{26 \pi}{7}\), first convert the angle to an equivalent angle between \(0\) and \(2\pi\) by finding the remainder when divided by \(2\pi\). Then, identify the quadrant based on the angle's position within the unit circle. Finally, calculate the reference angle by finding the difference between the angle and the nearest x-axis.

To find the equivalent angle of \(\frac{26 \pi}{7}\) within the interval \([0, 2\pi)\), we compute: \[ \text{angle\_mod} = \frac{26 \pi}{7} \mod (2\pi) \approx 5.3856 \]

The equivalent angle \(\text{angle\_mod} \approx 5.3856\) radians falls within the range: \[ 3\pi/2 < \text{angle\_mod} < 2\pi \] This indicates that the angle is in Quadrant IV.

The reference angle \(\theta_{\text{ref}}\) for an angle in Quadrant IV is given by: \[ \theta_{\text{ref}} = 2\pi - \text{angle\_mod} \approx 2\pi - 5.3856 \approx 0.8970 \]

Final Answer