Angles

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.

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

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.

\(315^\circ\) 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.

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.

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.

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.

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.

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

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 \(150^\circ\).

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.

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

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

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

(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.

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

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

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.

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.

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

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.

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

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 \(360^\circ\).

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.

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 \(360^\circ\).