Find the Angle and Length of the Arc in Different Quadrants of the Unit Circle

Given a unit circle centered at the origin, consider a point that makes an angle of \( \theta = \frac{7\pi}{6} \) radians with the positive x-axis. Find the quadrant in which this point lies and the length of the arc from the point where \( \theta = 0 \) to this point.

Solution:

1. Determine the quadrant:

The angle \( \theta = \frac{7\pi}{6} \) can be converted to degrees:

$ \theta = \frac{7\pi}{6} \times \frac{180}{\pi} = 210^{\circ} $

Since 210 degrees lies between 180 and 270 degrees, the point is in the third quadrant.

2. Calculate the length of the arc:

The length of an arc (s) in a unit circle is given by:

$ s = r \cdot \theta $

Because it is a unit circle (\( r = 1 \)):

$ s = 1 \cdot \frac{7\pi}{6} = \frac{7\pi}{6} $

Thus, the length of the arc is \( \frac{7\pi}{6} \) units.

Given a unit circle, find the quadrant where the angle ( heta = 5pi/4 ) radians lies and the corresponding arc length from the positive x-axis.

Solution:

1. Identify the quadrant:

The angle ( heta = frac{5pi}{4} ) in degrees is:

$ heta = frac{5pi}{4} imes frac{180}{pi} = 225^{circ} $

Since 225 degrees is between 180 and 270 degrees, the point lies in the third quadrant.

2. Determine the arc length:

In a unit circle:

$ s = r cdot heta $

For a unit circle (( r = 1 )):

$ s = 1 cdot frac{5pi}{4} = frac{5pi}{4} $

The arc length is ( frac{5pi}{4} ) units.

Find the quadrant and arc length for an angle of ( frac{11pi}{6} ) radians in a unit circle.

Solution:

1. Quadrant:

$ heta = frac{11pi}{6} imes frac{180}{pi} = 330^{circ} $

This is in the fourth quadrant.

2. Arc length:

$ s = 1 cdot frac{11pi}{6} = frac{11pi}{6} $

The arc length is ( frac{11pi}{6} ) units.

Start Using PopAi Today