In which quadrant does an angle of 5 radians terminate? Show your work.
I would say it is an angle whose coordinates (as a trigonometric angle measured along the unit circle, x^2+y^2=1), lie in the 3rd quadrant. Since one can see that pi < 3.5 < 3pi/2, where pi = 3.14 (approximately), the point which corresponds to this angle, 3.5 radians, on the unit circle must be in the 3rd quadrant.
So
3.5 radians = 3.5*180/π =200.53 Deg.
It lies in the 3rd Quadrant.
the question was about 5 radians.
3π/2 < 5 < 2π
so 5 radians is in QIV
To determine the quadrant in which an angle terminates, we need to consider the values of sine and cosine.
When an angle terminates in the first quadrant, both the sine and cosine values are positive. In the second quadrant, the sine value is positive but the cosine value is negative. In the third quadrant, both the sine and cosine values are negative. In the fourth quadrant, the sine value is negative but the cosine value is positive.
For an angle of 5 radians, we can find the values of sine and cosine using a calculator.
sin(5) ≈ 0.9589
cos(5) ≈ 0.2837
Since the sine value is positive (0.9589) and the cosine value is positive (0.2837), the angle terminates in the first quadrant.
Therefore, an angle of 5 radians terminates in the first quadrant.
To determine in which quadrant an angle of 5 radians terminates, we need to analyze the unit circle.
The unit circle is a circle with a radius of 1 unit centered at the origin (0, 0) of a Cartesian coordinate system. It contains all the possible terminal points of angles measured in radians.
One complete revolution around the unit circle is equal to 2π radians. Dividing the circle into quadrants, we have:
- Quadrant I: 0 radians to π/2 radians (∼1.5708 radians)
- Quadrant II: π/2 radians to π radians (∼3.1416 radians)
- Quadrant III: π radians to 3π/2 radians (∼4.7124 radians)
- Quadrant IV: 3π/2 radians to 2π radians (∼6.2832 radians)
To find the quadrant in which an angle of 5 radians terminates, we need to find the smallest quadrant that includes 5 radians.
Since 5 radians is between 4.7124 radians (3π/2) and 6.2832 radians (2π), it falls within the domain of Quadrant IV.
Therefore, an angle of 5 radians terminates in Quadrant IV.
Work:
1. Determine the range of each quadrant on the unit circle.
2. Compare 5 radians to the range of each quadrant.
3. Identify the smallest quadrant that includes 5 radians, which is Quadrant IV.