Question

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

16𝜋/3

Accepted Answer

Step 1: Understand that the angle is given in radians as $$\frac{16\pi}{3}. $$Since one full rotation around the circle is $$2\pi$$ radians, we need to find the equivalent angle between $$0$$ and $$2\pi by $$subtracting multiples of $$2\pi. $$Step 2: Calculate how many full rotations are contained in $$\frac{16\pi}{3} by $$dividing $$\frac{16\pi}{3} by$$ $$2\pi$$: $$\frac{16\pi}{3} \div 2\pi = \frac{16}{3} 	imes \frac{1}{2} = \frac{8}{3}. $$This means the angle makes $$2$$ full rotations (since $$\frac{8}{3} = 2 + \frac{2}{3}) $$plus an extra $$\frac{2}{3} of a $$rotation. Step 3: Find the remaining angle after subtracting $$2$$ full rotations: $$\frac{16\pi}{3} - 2 	imes 2\pi = \frac{16\pi}{3} - 4\pi = \frac{16\pi}{3} - \frac{12\pi}{3} = \frac{4\pi}{3}. $$This is the angle in standard position between $$0$$ and $$2\pi. $$Step 4: Determine the quadrant of the angle $$\frac{4\pi}{3}. $$Since $$\pi \u003c \frac{4\pi}{3} \u003c \frac{3\pi}{2}$$, the angle lies in the third quadrant. Step 5: To draw the angle, start from the positive x-axis and rotate counterclockwise by $$\frac{4\pi}{3}$$ radians, which is $$240^\circ ($$though we do not convert to degrees for the problem). Mark the terminal side of the angle in the third quadrant on the circle.

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

16𝜋/3

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Angles in Standard Position

Radian Measure and Circle Rotation

Quadrants of the Coordinate Plane

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.16𝜋/3

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Angles in Standard Position

Radian Measure and Circle Rotation

Quadrants of the Coordinate Plane

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

16𝜋/3