Multiple-choice questions on angles in standard position, involving both degrees and radians. Scroll down for step-by-step solutions.
An angle in standard position whose measure is \( -30^\circ \) has its terminal side in:
In which quadrant is the terminal side of an angle in standard position whose measure is \[ \frac{2\pi}{3} \]?
An angle in standard position whose measure is \( 1330^\circ \) has its terminal side in:
In which quadrant is the terminal side of an angle in standard position whose measure is \[ -\frac{7\pi}{4} \]?
An angle in standard position whose measure is \( -1550^\circ \) has its terminal side in:
In which quadrant is the terminal side of an angle in standard position whose measure is \[ -\frac{55\pi}{3} \]?
\( -30^\circ \)
Negative angles rotate clockwise. \[ -30^\circ = 360^\circ - 30^\circ = 330^\circ \] Since \( 330^\circ \) lies between \( 270^\circ \) and \( 360^\circ \), the terminal side is in Quadrant IV.
Answer: d) Quadrant IV
\( \frac{2\pi}{3} \)
Convert radians to degrees: \[ \frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ \] Since \( 120^\circ \) lies between \( 90^\circ \) and \( 180^\circ \), the angle is in Quadrant II.
Answer: b) Quadrant II
\( 1330^\circ \)
Subtract full rotations of \( 360^\circ \): \[ 1330^\circ - 3(360^\circ) = 250^\circ \] Since \( 250^\circ \) lies between \( 180^\circ \) and \( 270^\circ \), it is in Quadrant III.
Answer: c) Quadrant III
\( -\frac{7\pi}{4} \)
Add \( 2\pi \) to find a positive coterminal angle: \[ -\frac{7\pi}{4} + 2\pi = \frac{\pi}{4} \] Since \( \frac{\pi}{4} \) is between \( 0 \) and \( \frac{\pi}{2} \), it lies in Quadrant I.
Answer: a) Quadrant I
\( -1550^\circ \)
Add multiples of \( 360^\circ \): \[ -1550^\circ + 5(360^\circ) = 250^\circ \] Since \( 250^\circ \) is between \( 180^\circ \) and \( 270^\circ \), the terminal side is in Quadrant III.
Answer: c) Quadrant III
\( -\frac{55\pi}{3} \)
Add multiples of \( 2\pi \): \[ -\frac{55\pi}{3} + \frac{60\pi}{3} = \frac{5\pi}{3} \] Since \( \frac{5\pi}{3} \) is between \( \frac{3\pi}{2} \) and \( 2\pi \), it lies in Quadrant IV.
Answer: d) Quadrant IV