Angles

Correct answer: A. (300^\circ)

( \frac{5\pi}{3}\times \frac{180^\circ}{\pi}=300^\circ). Counting in (60^\circ) steps helps with standard angles.

Correct answer: C. (330^\circ)

( \frac{11\pi}{6} ) radians is (330^\circ). Take ( \frac{\pi}{6}=30^\circ) and multiply by (11).

Correct answer: C. Coterminal angles

(0^\circ) and (360^\circ) have the same terminal side so they are coterminal. Look for a difference of (360^\circ).

Correct answer: C. (405^\circ)

(45^\circ+360^\circ=405^\circ) so it is coterminal. Add or subtract (360^\circ) for coterminal angles.

Correct answer: B. −240°

Coterminal angles have the same terminal side and differ by an integer multiple of 360°. To obtain a negative angle from 120°, subtract one full revolution: 120° − 360° = −240°. Therefore option B is correct. Adding or subtracting 360° preserves the terminal side, so −240° is coterminal with 120°. Option A is positive and is not obtained by a 360° multiple from 120°; −120° differs by 240°, not 360°; and 480° is positive, although it is a positive coterminal angle because 120° + 360° = 480°. The word negative is essential.

Correct answer: C. \(330^\circ\)

Coterminal angles differ by an integral multiple of \(360^\circ\). Thus, \(-30^\circ+360^\circ=330^\circ\), so \(330^\circ\) is the positive coterminal angle. \(300^\circ\) has a different terminal side, so it is not coterminal with \(-30^\circ\). Exam tip: To obtain the least positive coterminal angle of a negative angle, add \(360^\circ\).

Correct answer: B. 30°

Coterminal angles have the same terminal side and differ by an integral multiple of 360°, because 360° represents one complete revolution. Since the required representative must lie between 0° and 360°, reduce 390° by one full revolution: 390° − 360° = 30°. The result is within the stated interval, so option B is correct. Adding or subtracting another 360° would produce 390° or −330°, which do not satisfy the requested interval. Option A and option C do not preserve the terminal side of 390°, while option D is not obtained by a valid reduction. Thus 30° is the unique suitable coterminal angle.

Correct answer: A. \(30^\circ\)

Coterminal angles differ by an integral multiple of \(360^\circ\). Here, \(750^\circ-2\times360^\circ=750^\circ-720^\circ=30^\circ\), so \(30^\circ\) is the required angle. \(60^\circ\) is not coterminal because \(750^\circ-60^\circ=690^\circ\), which is not a multiple of \(360^\circ\). Exam tip: subtract multiples of \(360^\circ\) to reduce an angle to the interval from \(0^\circ\) to \(360^\circ\).

Correct answer: C. Third quadrant

(200^\circ) lies between (180^\circ) and (270^\circ) so it is in the third quadrant. Remember the standard quadrant intervals.

Correct answer: D. Fourth quadrant

(310^\circ) lies between (270^\circ) and (360^\circ) so it is in the fourth quadrant. Angles after (270^\circ) lie in the fourth quadrant.

Correct answer: A. First quadrant

(25^\circ) lies between (0^\circ) and (90^\circ) so it is in the first quadrant. Small positive angles often lie in the first quadrant.

Correct answer: C. Positive y-axis

In standard position, an angle begins with its initial ray on the positive x-axis and is measured counterclockwise. A rotation of 90° is one quarter of a complete 360° revolution, so the terminal ray points vertically upward from the origin. The upward direction is the positive y-axis; therefore option C is correct. The positive x-axis corresponds to 0° or 360°, the negative x-axis to 180°, and the negative y-axis to 270°. The terminal ray at 90° lies exactly on an axis, not inside a quadrant. This standard-position sequence also rules out all three distractors.

Correct answer: B. Negative x-axis

The governing idea is the standard position of an angle. The initial side begins on the positive x-axis, and positive angles are measured counterclockwise. A rotation of 180° is a half-turn, so the terminal ray points exactly opposite to the initial ray. The direction opposite the positive x-axis is the negative x-axis; therefore option B is correct. The standard axis positions provide a useful check: 0° or 360° lies on the positive x-axis, 90° lies on the positive y-axis, 180° lies on the negative x-axis, and 270° lies on the negative y-axis. Because 180° is an exact multiple of 90°, the terminal side is on an axis, not inside any quadrant. The other options correspond to 90°, 270°, and 0° or 360° respectively.

Correct answer: D. Negative y-axis

In standard position, the initial side lies on the positive x-axis and the angle is measured counterclockwise. Moving through the quarter-turns gives the following sequence: at 90° the terminal side is on the positive y-axis, at 180° it is on the negative x-axis, and at 270° it is on the negative y-axis. Therefore option D is correct. The same conclusion follows from 270° = 3 × 90°, meaning the ray makes three quarter-turns from the positive x-axis. The four reference positions are positive x at 0° or 360°, positive y at 90°, negative x at 180°, and negative y at 270°. Since 270° is an exact multiple of 90°, the terminal side lies on an axis rather than within a quadrant.

Correct answer: B. ( \frac{180^\circ}{\pi} )

To convert radians to degrees multiply by ( \frac{180^\circ}{\pi} ). Remember the direction of the formula.

Correct answer: C. ( \frac{\pi}{180} )

To convert degrees to radians multiply by ( \frac{\pi}{180} ). The same rule applies to angles like (75^\circ).

Correct answer: D. 60'

In angular measurement, 1 degree is divided into 60 minutes; that is, \(1^\circ=60'\). Therefore, the correct answer is 60'. Note that 90' equals \(1.5^\circ\), so it is not correct. Exam tip: Remember \(1^\circ=60'\) and \(1'=60''\).

Correct answer: A. 60''

In angular measurement, 1 minute is divided into 60 seconds. Therefore, \(1' = 60''\), so 60'' is correct. 30'' represents half a minute, not one full minute. Exam tip: Remember that \(1^\circ = 60'\) and \(1' = 60''\).

Correct answer: C. \(120'\)

One degree contains \(60'\) (minutes). Therefore, \(2^\circ=2\times60'=120'\), so option C is correct. \(60'\) equals only \(1^\circ\), making it a close but incorrect option. Exam tip: To convert degrees to minutes, multiply the number of degrees by 60.

Correct answer: D. \(3^\circ\)

In angular measurement, \(1^\circ=60'\). Therefore, converting \(180'\) into degrees gives \(180\div60=3\), so \(180'=3^\circ\). The option \(4^\circ\) is incorrect because it equals \(240'\). Exam tip: divide minutes by 60 to convert them into degrees.

Correct answer: B. (15^\circ)

( \frac{\pi}{12}\times \frac{180^\circ}{\pi}=15^\circ). Dividing (180^\circ) by the denominator is an easy method.

Correct answer: B. ( \frac{5\pi}{4} ) radians

(225^\circ=\frac{225\pi}{180}=\frac{5\pi}{4}). Multiply by ( \frac{\pi}{180} ) to convert degrees to radians.

Correct answer: B. Second quadrant

(135^\circ) lies between (90^\circ) and (180^\circ) so it is in the second quadrant. Remember the interval to decide the position.

Correct answer: C. \(60^\circ\)

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(420^\circ-360^\circ=60^\circ\), the coterminal angle between \(0^\circ\) and \(360^\circ\) is \(60^\circ\). \(90^\circ\) is a different angle and cannot be obtained from \(420^\circ\) by adding or subtracting a whole multiple of \(360^\circ\). Exam tip: add or subtract \(360^\circ\) until the angle lies in the required interval.

Correct answer: D. 180'

One degree contains 60 minutes, i.e. \(1^\circ=60'\). Therefore, \(3^\circ=3\times60'=180'\). The nearby option 150' equals \(2.5^\circ\), not 3°. Exam tip: multiply degrees by 60 to convert them into minutes.