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Difficulty: HardUnit Circle and Angle Measures (Radians and Degrees)

For an angle in standard position on the unit circle, match each rotation scenario on the left with its corresponding terminal angle and location on the right.

  • A wheel rotates counterclockwise. Starting from the positive xx-axis, a point on the rim completes 3.753.75 full revolutions.3π2\frac{3\pi}{2} radians (terminal side on the negative yy-axis)
  • A terminal ray rotates counterclockwise by 5π3\frac{5\pi}{3} radians, and then rotates clockwise by 450∘450^\circ.7π6\frac{7\pi}{6} radians (terminal side in Quadrant III)
  • A point starts at (1,0)(1, 0) and travels a distance of 11π4\frac{11\pi}{4} units in the clockwise direction along the unit circle.5π4\frac{5\pi}{4} radians (terminal side in Quadrant III)
  • A terminal ray rotates clockwise by 150∘150^\circ and then counterclockwise by 11π6\frac{11\pi}{6} radians.π\pi radians (terminal side on the negative xx-axis)

Answer

Matching the scenarios: the wheel rotation matches 3π2\frac{3\pi}{2} radians on the negative yy-axis; the combined rotation of 5π3\frac{5\pi}{3} and −450∘-450^\circ matches 7π6\frac{7\pi}{6} radians in Quadrant III; the clockwise travel of 11π4\frac{11\pi}{4} units matches 5π4\frac{5\pi}{4} radians in Quadrant III; the combined rotation of −150∘-150^\circ and 11π6\frac{11\pi}{6} matches π\pi radians on the negative xx-axis.
Each scenario is correctly matched by converting all angular values to radians, determining the net rotation direction (positive for counterclockwise, negative for clockwise), and finding the coterminal angle in the range [0,2π)[0, 2\pi) to locate the terminal side.

Step-by-Step Solution

1
Convert the rotation from revolutions to radians for the first scenario.
3.75 revolutions×2π radians/revolution=7.5π3.75 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 7.5\pi radians. Subtract 33 full rotations (6π6\pi radians) to find the coterminal angle in [0,2π)[0, 2\pi): 7.5π−6π=1.5π=3π27.5\pi - 6\pi = 1.5\pi = \frac{3\pi}{2} radians. This lies on the negative yy-axis.
One full revolution corresponds to 2π2\pi radians, and subtracting multiples of 2π2\pi yields the coterminal position.
2
Calculate the net angle in radians for the second scenario.
Convert 450∘450^\circ to radians: −450∘×π180∘=−5π2-450^\circ \times \frac{\pi}{180^\circ} = -\frac{5\pi}{2} radians (negative due to clockwise direction). Net angle is 5π3−5π2=−5π6\frac{5\pi}{3} - \frac{5\pi}{2} = -\frac{5\pi}{6} radians. Find the positive coterminal angle: −5π6+2π=7π6-\frac{5\pi}{6} + 2\pi = \frac{7\pi}{6} radians. This lies in Quadrant III.
Converting all angles to radians with correct sign conventions allows addition to find the net angle.
3
Relate arc length to angle measure on the unit circle for the third scenario.
On a circle with r=1r = 1, the arc length s=11π4s = \frac{11\pi}{4} corresponds to a rotation of 11π4\frac{11\pi}{4} radians. Clockwise direction makes it −11π4-\frac{11\pi}{4} radians. Find the coterminal angle in [0,2π)[0, 2\pi): −11π4+4π=5π4-\frac{11\pi}{4} + 4\pi = \frac{5\pi}{4} radians. This lies in Quadrant III.
The arc length formula s=rθs = r\theta simplifies to s=θs = \theta on the unit circle, and clockwise motion represents a negative angle.
4
Compute the net angle in radians for the fourth scenario.
Convert −150∘-150^\circ to radians: −150∘×π180∘=−5π6-150^\circ \times \frac{\pi}{180^\circ} = -\frac{5\pi}{6} radians. Net angle is −5π6+11π6=6π6=π-\frac{5\pi}{6} + \frac{11\pi}{6} = \frac{6\pi}{6} = \pi radians. This lies on the negative xx-axis.
Converting degrees to radians enables direct fraction addition to determine the final terminal position.

Key Concept

Calculating coterminal angles and conversions between degrees, radians, and revolutions on the unit circle.
Estimated Time:2m 30s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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