Angles

Correct answer: B. \(2\) revolutions

One complete revolution is \(360^\circ\). So the number of revolutions is \(\dfrac{720^\circ}{360^\circ}=2\). Therefore, \(720^\circ\) equals \(2\) complete revolutions.

Correct answer: B. (3)

One half revolution is (180^\circ), and (540^\circ\div180^\circ=3). Divide by (180^\circ) for half revolutions.

Correct answer: C. 60'

In angular measurement, one degree is divided into 60 equal parts, and each part is called one minute. Therefore, \(1^\circ=60'\). Since \(30'\) equals half a degree, it is not correct. Exam tip: remember \(1^\circ=60'\) and \(1'=60''\) for degree-minute-second conversions.

Correct answer: C. \(315^\circ\)

\(315^\circ\) lies between \(270^\circ\) and \(360^\circ\), so its terminal side is in the fourth quadrant. \(225^\circ\) lies in the third quadrant. Exam tip: place an angle between \(0^\circ\) and \(360^\circ\) before identifying its quadrant.

Correct answer: A. (\frac{1}{2}^\circ)

The direct answer is option A, \(\frac{1}{2}^{\circ}\). An angle of one degree is divided into 60 minutes, so \(1^{\circ}=60'\). Therefore one minute is \(1'=\frac{1}{60}^{\circ}\). For 30 minutes, multiply by 30: \(30'=30\times\frac{1}{60}^{\circ}=\frac{30}{60}^{\circ}=\frac{1}{2}^{\circ}\). Option A is correct. Option B, \(1^{\circ}\), represents 60 minutes, not 30. Option C, \(2^{\circ}\), represents 120 minutes. Option D, \(\frac{1}{3}^{\circ}\), would equal 20 minutes, since \(60\times\frac{1}{3}=20\). Remember that the prime mark denotes minutes, not multiplication. Memory cue: convert angle minutes to degrees by dividing by 60.

Correct answer: B. \(\frac{3}{2}'\)

Since \(1'=60''\), convert \(90''\) into minutes by dividing by 60: \(90''\div 60=\frac{3}{2}'\). Therefore, \(\frac{3}{2}'\) is correct. Note that \(\frac{1}{2}'\) equals only \(30''\). Exam tip: divide by 60 when converting seconds to minutes.

Correct answer: B. π/12

For degree-to-radian conversion, multiply the degree measure by π/180°. Therefore, 15° × π/180° = (15π/180) radians. Dividing numerator and denominator by 15 gives π/12 radians, so option B is correct. A useful check is that 30° equals π/6; 15° is half of 30°, so its radian measure must be π/12. The distractors represent other standard angles: π/6 is 30°, π/10 is 18°, and π/8 is 22.5°. Keeping the degree symbol in the conversion factor conceptually shows that the degree units cancel and the result is expressed in radians.

Correct answer: C. \(405^\circ\)

Coterminal angles have the same initial and terminal sides, so their measures differ by a multiple of \(360^\circ\). Since \(405^\circ-45^\circ=360^\circ\), option C is correct. Exam tip: check the difference first.

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

(225^\circ \times \frac{\pi}{180^\circ}=\frac{5\pi}{4}). Remembering (45^\circ=\frac{\pi}{4}) also helps.

Correct answer: B. \(150^\circ\)

Use \(\pi\) radians \(=180^\circ\). So \(\dfrac{5\pi}{6}=\dfrac{5}{6}\pi\) corresponds to \(\dfrac{5}{6}\times 180^\circ=150^\circ\). Hence, \(\dfrac{5\pi}{6}\) radians equals \(150^\circ\).

Correct answer: C. (210^\circ)

Radians and degrees measure the same angle in different units. The basic conversion is \\(\pi\\) radians = \\(180^\circ\\). Therefore, to convert a radian measure into degrees, multiply it by \\(\frac{180^\circ}{\pi}\\). The factor \\(\pi\\) cancels, leaving a degree measure.

For this angle, \\(\frac{7\pi}{6}\times\frac{180^\circ}{\pi}=7\times30^\circ=210^\circ\\). Equivalently, \\(\frac{\pi}{6}=30^\circ\\), so seven such parts make \\(210^\circ\\). Thus option C is correct. The angle is greater than 180 degrees but less than 360 degrees, which is consistent with the result.

Correct answer: D. (240^\circ)

(\frac{4\pi}{3}\times\frac{180^\circ}{\pi}=240^\circ). Since (\frac{\pi}{3}=60^\circ), four times it is (240^\circ).

Correct answer: C. Third quadrant

(210^\circ) lies between (180^\circ) and (270^\circ), so it is in the third quadrant. Remember the boundary angles.

Correct answer: D. Fourth quadrant

(315^\circ) lies between (270^\circ) and (360^\circ), so it is in the fourth quadrant. Decide the position using boundaries.

Correct answer: B. Second quadrant

(100^\circ) lies between (90^\circ) and (180^\circ), so it is in the second quadrant. After (90^\circ) and before (180^\circ) is the second quadrant.

Correct answer: A. First quadrant

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

Correct answer: B. (240^\circ)

(-120^\circ+360^\circ=240^\circ). Add (360^\circ) to bring a negative angle into the principal range.

Correct answer: B. \(45^\circ\)

To find a coterminal angle, subtract multiples of \(360^\circ\) from the given angle. \(765^\circ-2\times360^\circ=765^\circ-720^\circ=45^\circ\). Hence, \(45^\circ\) is the coterminal angle of \(765^\circ\) between \(0^\circ\) and \(360^\circ\). Although \(35^\circ\) is a nearby option, it does not differ from \(765^\circ\) by a whole multiple of \(360^\circ\). Exam tip: reducing an angle modulo \(360^\circ\) gives its coterminal angle in this interval.

Correct answer: C. 270^\circ

Coterminal angles differ by an integral multiple of 360^\circ. \((-450^\circ)+2(360^\circ)=270^\circ\), and this lies between 0^\circ and 360^\circ. Obtaining 90^\circ would require adding 540^\circ, which is not a multiple of 360^\circ, so it is not coterminal with the given angle. Exam tip: For a negative angle, keep adding 360^\circ until it lies in the required interval.

Correct answer: B. (45^\circ)

(\frac{\pi}{4}\times\frac{180^\circ}{\pi}=45^\circ). After canceling (\pi), calculate (180\div4).

Correct answer: A. \(30^\circ,\ 390^\circ\)

Coterminal angles have the same initial and terminal sides, so their difference must be an integral multiple of \(360^\circ\). Here, \(390^\circ-30^\circ=360^\circ\). Exam tip: subtract the angles first.

Correct answer: A. The angle with no rotation

An angle with no rotation is called a zero angle. In definition questions identify the key word.

Correct answer: B. \(180^\circ\)

A straight angle measures \(180^\circ\). Its two arms extend in opposite directions and form a straight line, so option B is correct. \(90^\circ\) is a right angle, not a straight angle. Exam tip: remember that a right angle is \(90^\circ\), a straight angle is \(180^\circ\), and a complete angle is \(360^\circ\).

Correct answer: C. 90°

An angle measuring 90° is called a right angle. The angle formed between two perpendicular lines is 90°. In contrast, 60° is an acute angle because it is less than 90°. Exam tip: remember that 90° is a right angle, 180° is a straight angle, and 360° is a complete angle.

Correct answer: D. \(360^\circ\)

A complete angle represents one full revolution. One complete turn measures \(360^\circ\), so \(360^\circ\) is correct. An angle of \(180^\circ\) is called a straight angle, not a complete angle. Exam tip: Remember that a right angle is \(90^\circ\), a straight angle is \(180^\circ\), and a complete angle is \(360^\circ\).