Determine the quadrant in which an angle, θ, lies if θ = 8π / 7
a. 3rd quadrant
b. 1st quadrant
c. 4th quadrant
d. 2nd quadrant
8π / 7= π + (1/7)π
so just a bit more than π,
so in quadrant III
here is a "sneaky way"
set your calculator to radians
take sin 8π/7 to get a negative
take cos 8π/7 to get a negative
the only quadrant in which both sine and cosine are negative is III
To determine the quadrant in which an angle θ lies, we need to examine the value of θ.
In this case, θ = 8π / 7.
To convert this angle to degrees, we can use the conversion factor of 180 degrees / π radians:
θ (in degrees) = (8π / 7) * (180 degrees / π) = 102.857 degrees
Now, think about the four quadrants on the coordinate plane.
- The 1st quadrant is between 0 and 90 degrees.
- The 2nd quadrant is between 90 and 180 degrees.
- The 3rd quadrant is between 180 and 270 degrees.
- The 4th quadrant is between 270 and 360 degrees.
Since θ = 102.857 degrees, it lies in the 2nd quadrant.
Therefore, the correct answer is d. 2nd quadrant.
To determine the quadrant in which an angle θ lies, you need to examine the value of θ. Here's how to do it:
1. Recall that one full revolution in radians is equal to 2π. In other words, an angle of 2π or 360 degrees represents a complete circle.
2. Divide the given angle θ by 2π: θ / 2π.
3. Simplify the fraction if possible. In this case, θ = 8π / 7.
θ / 2π = (8π / 7) / (2π) = 8π / (7 * 2π) = 8 / 14 = 4 / 7.
4. Once you have simplified the fraction, examine the resulting value. The numerator tells you the number of complete revolutions, and the denominator represents the parts of a revolution that remain.
In this case, 4 / 7 indicates that there are four complete revolutions (4 times around the unit circle) and a remaining fraction of 7 parts.
5. Determine the quadrant based on the remaining fraction:
a. If the remaining fraction is less than 1/4 or 90° (π/2), the angle lies in the 1st quadrant.
b. If the remaining fraction is between 1/4 and 1/2 or 90° (π/2) and 180° (π), the angle lies in the 2nd quadrant.
c. If the remaining fraction is between 1/2 and 3/4 or 180° (π) and 270° (3π/2), the angle lies in the 3rd quadrant.
d. If the remaining fraction is greater than 3/4 or 270° (3π/2), the angle lies in the 4th quadrant.
6. In this case, the remaining fraction is 4/7, which is greater than 3/4 or π/2. Therefore, the angle θ = 8π / 7 lies in the 4th quadrant.
So, the answer is: c. 4th quadrant.