Angles

In standard position, the initial side lies on the positive x-axis. Moving anticlockwise, \(90^\circ\) reaches the positive y-axis and \(180^\circ\) the negative x-axis; hence \(270^\circ\) lies on the negative y-axis. Exam tip: remember the axis order every \(90^\circ\).

(90^\circ) is half of (180^\circ) so it is ( \frac{\pi}{2} ). Divide the degree measure by (180^\circ) first.

Coterminal angles have the same initial and terminal sides, and their difference is an integral multiple of \(360^\circ\). Since \(30^\circ+360^\circ=390^\circ\), A is correct. The difference for \(330^\circ\) is \(300^\circ\), not \(360^\circ\). Exam tip: use \(\theta\pm360^\circ\).

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

( \pi ) radians equals (180^\circ). Multiply by ( \frac{180^\circ}{\pi} ) to convert radians to degrees.

To convert radians into degrees, multiply by \(\frac{180^\circ}{\pi}\): \(\frac{\pi}{3}\times\frac{180^\circ}{\pi}=60^\circ\). Hence, \(60^\circ\) is correct. \(30^\circ\) is a close distractor because its radian measure is \(\frac{\pi}{6}\), not \(\frac{\pi}{3}\). Exam tip: Remember that \(\pi\) radians = \(180^\circ\).

( \frac{\pi}{6}\times \frac{180^\circ}{\pi}=30^\circ). For radians to degrees always use ( \frac{180^\circ}{\pi} ).

The answer is option D, 45 degrees. Radians and degrees measure the same angle in different units. The basic conversion fact is pi radians=180 degrees. To convert pi/4 radians, divide both sides of the basic relation by 4: pi/4 radians=180/4 degrees=45 degrees. Option A, 30 degrees, would correspond to pi/6 radians, not pi/4. Option B, 60 degrees, corresponds to pi/3 radians. Option C, 90 degrees, corresponds to pi/2 radians. Option D, 45 degrees, exactly matches pi/4, so it is correct. A quick way to remember this is that a right angle is pi/2 radians or 90 degrees, and half of a right angle is pi/4 radians or 45 degrees. Do not confuse the denominator 4 with 4 degrees; the pi-to-180 conversion must be used.

A positive angle is conventionally measured from the positive x-axis in the anticlockwise direction. This convention is used in trigonometry and coordinate geometry. Moving anticlockwise corresponds to increasing the angle measure, while moving clockwise corresponds to a negative angle when no other convention is specified.

For example, starting at the positive x-axis, a quarter turn anticlockwise is \\(90^\circ\\), a half turn is \\(180^\circ\\), and a full turn is \\(360^\circ\\). These standard measurements show why the positive direction is anticlockwise. Therefore option A is correct. The other choices describe vertical movement or the opposite rotational direction, not the usual angular convention.

In standard position, an angle is measured from its initial arm. Angles measured anticlockwise are positive, whereas angles measured clockwise are negative. Therefore, the correct answer is clockwise direction. Anticlockwise is the closest distractor because it represents positive angles. Exam tip: remember the sign convention—anticlockwise (+), clockwise (−).

One complete revolution is (360^\circ). Adding or subtracting (360^\circ) helps in coterminal angles.

A straight angle forms a straight line, so its measure is \(180^\circ\). An angle of \(90^\circ\) is a right angle, while \(360^\circ\) is a complete angle. Exam tip: whenever an angle forms a straight line, use \(180^\circ\).

The governing conversion is 180° = π radians. A right angle measures 90°, which is exactly half of a straight angle measuring 180°. Consequently, its radian measure is half of π: 90 × π/180 = π/2 radians. Therefore option B is correct. The alternatives correspond to other standard angles. π radians represents 180°, a straight angle; π/3 radians represents 60°; and 2π radians represents 360°, one complete revolution. A reliable way to check the result is to remember the benchmark values 90° = π/2, 180° = π, and 360° = 2π. Since a right angle is one quarter of a full revolution, it must also equal one quarter of 2π, which again gives π/2.

(360^\circ) is two times (180^\circ) so it is (2\pi) radians. Remember the radian measure of a full turn.

To convert a degree measure into radians, use radians = degrees × π/180. For 270°, the calculation is 270 × π/180. Reducing the numerical fraction by 90 gives 3/2, so 270° = 3π/2 radians. Hence option A is correct. The result can also be verified geometrically: 270° contains three right angles, and each right angle equals π/2 radians; therefore 3 × π/2 = 3π/2. Option B is π/2, the measure of 90°. Option C, 2π/3, equals 120°, and option D, 5π/4, equals 225°. Comparing the degree size with 180° also helps: 270° is one and a half straight angles, so its radian measure should be 1.5π, not a value smaller than π.

To convert degrees to radians, multiply the angle by \(\frac{\pi}{180}\). Thus, \(120^\circ=120\times\frac{\pi}{180}=\frac{2\pi}{3}\) radians. Note that \(\frac{5\pi}{6}\) radians equals \(150^\circ\), so it is not correct here. Exam tip: Remember that \(180^\circ=\pi\) radians.

The governing conversion factor is π/180 radians per degree. Therefore, 210° = 210 × π/180 radians. Simplify the numerical fraction by dividing 210 and 180 by 30: 210/180 = 7/6. Thus the answer is 7π/6 radians, so option C is correct. The distractors represent nearby but different standard angles: 5π/6 radians equals 150°, 3π/2 radians equals 270°, and 4π/3 radians equals 240°. Another reasonableness check is useful: 210° is greater than 180° but less than 240°, so its radian measure must be greater than π but less than 4π/3. The value 7π/6 lies in precisely that interval, confirming the calculation and the selected option.

Angles in the third quadrant are greater than \(180^\circ\) and less than \(270^\circ\). Since \(210^\circ\) lies in this interval, it is correct. \(135^\circ\) is in quadrant II. Exam tip: check the interval first.

The answer is option A, 5pi/3 radians. To convert degrees into radians, multiply the degree measure by pi/180. Therefore 300 degrees=(300pi)/180 radians. Simplifying by dividing numerator and denominator by 60 gives 5pi/3 radians. Option A has exactly this value, so it is correct. Option B, 4pi/3, equals 240 degrees because 4 times 180 divided by 3 is 240. Option C, 7pi/4, equals 315 degrees, not 300 degrees. Option D, 3pi/2, equals 270 degrees. Another useful check is that 300 degrees is 60 degrees short of 360 degrees, so its radian measure is 2pi-pi/3=5pi/3. The answer must be less than 2pi because 300 degrees is less than a full turn. Remember the conversion direction: degrees to radians means multiply by pi/180.

\(45^\circ\) is greater than \(0^\circ\) and less than \(90^\circ\), so it is an acute angle. \(90^\circ\) is a right angle and is not in this open interval. Exam tip: check whether endpoint values are included.

(330^\circ=\frac{330\pi}{180}=\frac{11\pi}{6}). Multiples of (30^\circ) often give denominator (6).

( \frac{2\pi}{3}\times \frac{180^\circ}{\pi}=120^\circ). Cancel ( \pi ) first and then multiply.

For conversion from radians to degrees, use π radians = 180°, or multiply the radian measure by 180°/π. Thus (3π/4) × (180°/π) is the required calculation. The π factors cancel, leaving (3 × 180°)/4 = 540°/4 = 135°. Therefore option A is correct. A useful geometric check is that 3π/4 is three-fourths of π, and π radians represents a straight angle of 180°; three-fourths of 180° is 135°. Option B, 120°, corresponds to 2π/3 radians. Option C, 150°, corresponds to 5π/6 radians, while option D, 180°, corresponds to π radians. Cancellation of π before simplifying avoids confusing this value with nearby standard angles.

Radians and degrees are two ways of measuring the same angle. The basic conversion fact is \(\pi\) radians equal \(180^\circ\), so \(\frac{\pi}{6}\) radians equals \(30^\circ\). The numerator 5 means that this basic sixth-angle is taken five times. This direct relationship is more reliable than trying to estimate the angle from the decimal value of \(\pi\).

Use \(\frac{5\pi}{6}\times\frac{180^\circ}{\pi}\). The \(\pi\) factors cancel, leaving \(\frac{5\times180^\circ}{6}=5\times30^\circ=150^\circ\). Hence option B is correct. The values \(120^\circ\), \(210^\circ\), and \(300^\circ\) correspond to different multiples of \(\pi/6\), not to five such parts.

( \frac{7\pi}{6}\times \frac{180^\circ}{\pi}=210^\circ). Multiply by (180^\circ) while converting radians to degrees.