Angles

In standard position the initial side lies on the positive (x)-axis. In exams identify the initial and terminal sides separately.

By the standard convention, rotation anticlockwise from the initial arm is taken as positive, so the angle formed is a positive angle. Clockwise rotation represents a negative angle, which is the closest distractor. Exam tip: check the direction of rotation before deciding the sign of an angle.

The governing convention for directed angles is that counterclockwise rotation is positive and clockwise rotation is negative. This sign describes the direction of rotation, not the size of the angle. Thus, any clockwise turn—whether it is 30°, 90°, 180°, or more—is represented by a negative directed angle, so option B is correct. An acute angle is determined by a magnitude below 90°, and an obtuse angle by a magnitude between 90° and 180°; neither description tells us the sign of the rotation. Option A applies to counterclockwise motion under the standard convention. Therefore the direction word “clockwise” directly identifies a negative angle.

One complete revolution means returning to the starting position after turning once around a point, so its measure is 360°. In contrast, 180° represents a half revolution and 90° represents a quarter revolution. Exam tip: To convert revolutions into degrees, multiply the number of revolutions by 360°.

One complete revolution measures 360°. Therefore, half a revolution is \(\frac{360°}{2}=180°\). In contrast, 90° is a quarter revolution, while 360° is a full revolution. Exam tip: Multiply the fraction of a revolution by 360° to find the angle.

One complete revolution around a point measures 360°. A quarter revolution means that the full turn is divided into four equal parts. Therefore, its measure is calculated as 360° ÷ 4 = 90°. Hence option A is correct. The value 180° represents a half revolution, 270° represents three quarters of a revolution, and 120° would represent one third of a full turn, not one quarter. This result is also consistent with the geometry of a right angle: two perpendicular rays divide a full circular turn into four equal quarter-turns, each measuring 90°. The calculation and the geometric interpretation agree.

The fundamental conversion relationship between degrees and radians is 180° = π radians. It follows from the definition of a radian: in a circle, a complete revolution is 2π radians and also 360°, so half a revolution is π radians and 180°. Therefore, 180° corresponds exactly to π radians, making option C correct. Option D, π/2, corresponds to 90°, while option B, 2π, corresponds to a complete 360° revolution. The number 1 is not the radian measure of a straight angle. Remembering the half-turn relationship is enough to convert many standard angles accurately.

Since (180^\circ=\pi), (360^\circ=2\pi). When the angle doubles the radian measure also doubles.

(90^\circ) is half of (180^\circ), so its radian measure is (\frac{\pi}{2}). Half the angle means half the radian measure.

(45^\circ) is (\frac{1}{4}) of (180^\circ), so it is (\frac{\pi}{4}) radians. The ratio method is simple.

(30^\circ \times \frac{\pi}{180^\circ}=\frac{\pi}{6}). Multiply by (\frac{\pi}{180^\circ}) to convert degrees to radians.

An angle in the second quadrant is greater than \(90^\circ\) and less than \(180^\circ\). Since \(120^\circ\) lies in this interval, it is correct. \(210^\circ\) is in the third quadrant. Exam tip: remember the quadrant boundaries \(90^\circ,180^\circ,270^\circ\).

(120^\circ \times \frac{\pi}{180^\circ}=\frac{2\pi}{3}). Reduce the fraction before writing the answer.

The governing conversion is \(180^\circ=\pi\) radians. Therefore, to convert degrees to radians, multiply the degree measure by \(\frac{\pi}{180^\circ}\). For 150°, \(150\times\frac{\pi}{180}=\frac{150\pi}{180}=\frac{5\pi}{6}\), after dividing numerator and denominator by 30. Hence option A is correct. The alternatives represent different standard angles: \(\frac{3\pi}{4}=135^\circ\), \(\frac{2\pi}{3}=120^\circ\), and \(\frac{7\pi}{6}=210^\circ\). A reasonableness check also helps: 150° lies between 90° and 180°, so its radian value must lie between \(\frac\pi2\) and \(\pi\); \(\frac{5\pi}{6}\) satisfies that condition.

(\frac{\pi}{3} \times \frac{180^\circ}{\pi}=60^\circ). Multiply by (\frac{180^\circ}{\pi}) to convert radians to degrees.

The governing equivalence is \(\pi\) radians = 180°. To convert \(\frac\pi6\) radians into degrees, multiply by \(\frac{180^\circ}{\pi}\): \(\frac\pi6\times\frac{180^\circ}{\pi}=\frac{180^\circ}{6}=30^\circ\). The factors \(\pi\) cancel, so option C is correct. The distractors correspond to nearby standard conversions: 60° is \(\frac\pi3\) radians, 45° is \(\frac\pi4\) radians, and 15° is \(\frac\pi{12}\) radians. Writing the units in the conversion factor prevents confusion and shows why the result is in degrees. This method is general: multiply any radian measure by \(180^\circ/\pi\) to obtain its degree measure.

Since \(\pi\) radians = \(180^\circ\), \(\frac{\pi}{2}\) radians = \(\frac{180^\circ}{2}=90^\circ\). Thus, it is a right angle and one-quarter of a complete revolution. \(45^\circ\) corresponds to \(\frac{\pi}{4}\) radians. Exam tip: multiply radians by \(\frac{180^\circ}{\pi}\) to convert them into degrees.

Use the standard equivalence \(\pi\) radians = 180°. The degree measure of \(\frac{2\pi}{3}\) is found by multiplying by \(\frac{180^\circ}{\pi}\): \(\frac{2\pi}{3}\times\frac{180^\circ}{\pi}=\frac{2\times180^\circ}{3}=120^\circ\). Therefore option A is correct. A quick check using standard angles gives the same result: \(\frac\pi3=60^\circ\), so twice that angle, \(\frac{2\pi}{3}\), is 120°. The other choices represent different radian values: 135° is \(\frac{3\pi}{4}\), 150° is \(\frac{5\pi}{6}\), and 180° is \(\pi\). Cancelling \(\pi\) before simplifying keeps the conversion accurate.

(\frac{3\pi}{2}\times\frac{180^\circ}{\pi}=270^\circ). The numerator (3) makes the final value three times (90^\circ).

(270^\circ=\frac{270\pi}{180}=\frac{3\pi}{2}). You can also think of (90^\circ) as (\frac{\pi}{2}).

The difference between coterminal angles is an integral multiple of (360^\circ). Use this rule when the terminal side is the same.

Coterminal angles have the same initial and terminal sides, so their difference is an integral multiple of \(360^\circ\). Since \(30^\circ+360^\circ=390^\circ\), \(390^\circ\) is a positive coterminal angle of \(30^\circ\). The difference between \(300^\circ\) and \(30^\circ\) is \(270^\circ\), which is not a multiple of \(360^\circ\). Exam tip: use \(\theta+360^\circ n\), where \(n\) is an integer, to find coterminal angles.

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(45^\circ-360^\circ=-315^\circ\), \(-315^\circ\) is a negative coterminal angle of \(45^\circ\). The angle \(-45^\circ\) differs from \(45^\circ\) by \(90^\circ\), so it is not coterminal. Exam tip: subtract \(360^\circ\) from a given angle to obtain a negative coterminal angle.

Coterminal angles differ by an integer multiple of one full revolution, and one full revolution is 360°. To place 400° in the interval from 0° to 360°, subtract 360° once: 400° − 360° = 40°. Since 40° lies within the required interval, it is the required coterminal angle, so option C is correct. Adding or subtracting 360° preserves the terminal side of an angle. The values 20°, 30°, and 50° do not differ from 400° by a whole multiple of 360°, so they are not coterminal with it. Because 400° is only one full turn plus 40°, the answer is immediate after one subtraction.

Coterminal angles differ by an integral multiple of 360°. Here, \((-30^\circ+360^\circ)=330^\circ\), which lies between 0° and 360°. Although 300° is a nearby distractor, its difference from \(-30^\circ\) is 330°, not a multiple of 360°. Exam tip: To convert a negative angle into the 0° to 360° range, first add 360°.