19pi/7

Which quadrant does the terminal side lie on?

19 pi / 7 =

( 14 + 5 ) * pi / 7 =

( 14 / 7 ) pi + ( 5 / 7 ) pi =

2 pi + ( 5 / 7 ) pi

2 pi + theta = theta

I quadrant 0 ÷ pi / 2

II quadrant pi / 2 ÷ pi

III quadrant pi ÷ ( 3 / 2 ) pi

IV quadrant ( 3 / 2 ) pi ÷ 2 pi

theta = ( 5 / 7 ) pi

theta = 0.7142857 pi

thheta > pi /2

theta < pi

II quadrant

Some students can see this better when visualizing it in degrees

19π/7 radians = 19(180)/7 ° = 488.57..°
= co-terminal with 488.57-360 or 128.57..°

which would be in quadrant II

To determine the quadrant in which the terminal side lies, we need to examine the value of the angle 19π/7.

First, let's convert 19π/7 to a decimal approximation.
19π/7 ≈ 8.6394

Since the value of the angle is positive and greater than 2π (or 360 degrees), we can conclude that the terminal side completes more than one full revolution around the origin.

To determine the quadrant, we can subtract 2π from the angle until we obtain a value within the range of 0 to 2π.

8.6394 - 2π ≈ 8.6394 - 6.283 ≈ 2.3564

The value 2.3564, which is in the second quadrant, tells us that the terminal side lies in the second quadrant.

Therefore, the terminal side of the angle 19π/7 lies in the second quadrant.

To determine which quadrant the terminal side of the angle lies on, we need to consider the value of the angle.

To find the quadrant, we can use the following steps:

1. Divide the angle by 2π (360 degrees) to normalize it to a range between 0 and 2π (0 and 360 degrees).

In this case, we have (19π/7)/(2π) = 19/7 ≈ 2.7143.

2. Reduce the normalized value to an equivalent value between 0 and 4, where 0 represents the positive x-axis.

Since 2.7143 is greater than 2, we can subtract 2 to get the reduced value of 0.7143.

3. Interpret the reduced value to determine the quadrant.

If the reduced value is between 0 and 1, it lies in the first quadrant (0 < θ < π/2 or 0 < θ < 90 degrees).
If the reduced value is between 1 and 2, it lies in the second quadrant (π/2 < θ < π or 90 < θ < 180 degrees).
If the reduced value is between 2 and 3, it lies in the third quadrant (π < θ < 3π/2 or 180 < θ < 270 degrees).
If the reduced value is between 3 and 4, it lies in the fourth quadrant (3π/2 < θ < 2π or 270 < θ < 360 degrees).

In our case, the reduced value is 0.7143, which is between 0 and 1. Therefore, the terminal side of 19π/7 lies in the first quadrant.