Angles

When arc length equals radius the angle is (1) radian. Remember the basic definition of radian.

The direct answer is option B: arc length. In the radian-measure formula \\(\\theta=s/r\\), theta is the angle measured in radians, s is the length of the arc cut off by the angle, and r is the radius of the circle. The relation comes from the definition of a radian: an angle of one radian subtends an arc whose length equals the radius. Therefore, rearranging the relation gives arc length s = r theta, and dividing by r gives \\(\\theta=s/r\\). Option A, radius, is represented by r, not s. Option B, arc length, is exactly what s represents, so it is correct. Option C, diameter, is twice the radius and is not the symbol s in this formula. Option D, area, is a different quantity; sector area involves a different formula, not simply s/r. Always check the units: s and r are both lengths, so their ratio is dimensionless, as a radian measure should be. Remember: s stands for “space along the circle” or arc length, while r stands for radius.

In this formula (r) represents the radius of the circle. In radian questions distinguish radius and arc length.

For an angle measured in radians, \(\theta=\frac{s}{r}\), where \(s\) is the arc length and \(r\) is the radius. Thus, \(\theta=\frac{10}{5}=2\) radians. It would be \(1\) radian only if the arc length were equal to the radius. Exam tip: Ensure that \(s\) and \(r\) are in the same unit before applying the formula.

Angles in the third quadrant are greater than 180° and less than 270°. Since 210° lies in this interval, its terminal side is in the third quadrant. In contrast, 315° lies in the fourth quadrant. Exam tip: compare the angle with 90° intervals.

For a central angle measured in radians, the arc-length formula is \(s=r\theta\). Thus, \(s=7\times4=28\) cm. The value 14 cm is only twice the radius; arc length must also be multiplied by the angle \(\theta\). Exam tip: use \(s=r\theta\) directly only when \(\theta\) is in radians.

The governing concept is the radian measure of an angle. For a sector, the arc length formula is s = rθ when θ is measured in radians. To find the radius, rearrange the relation as r = s/θ. Substituting the given values gives r = 16/2 = 8 cm. Therefore the radius is 8 cm, so option C is correct. The unit remains centimetres because arc length is measured in centimetres and the radian is dimensionless. Option A would result from dividing by 4 without justification, option B from an incorrect subtraction or estimate, and option D from multiplying s by θ instead of dividing. The key is to use radians, not degrees, in the formula.

To convert degrees into radians multiply by (\frac{\pi}{180^\circ}). If conversion direction reverses the factor also reverses.

To convert radians into degrees multiply by (\frac{180^\circ}{\pi}). Cancel (\pi) first and then calculate.

Angles in the second quadrant are greater than \(90^\circ\) and less than \(180^\circ\). Since \(120^\circ\) lies in this interval, it is correct. \(210^\circ\) lies in the third quadrant. Exam tip: use multiples of \(90^\circ\) as quadrant boundaries.

(105^\circ\times\frac{\pi}{180^\circ}=\frac{7\pi}{12}). Simplifying the ratio is the main step.

Coterminal angles have the same initial and terminal sides, and their difference is a whole multiple of \(360^\circ\). Here, \(405^\circ-45^\circ=360^\circ\). In exams, subtract the angles and check for a multiple of \(360^\circ\).

(330^\circ\times\frac{\pi}{180^\circ}=\frac{11\pi}{6}). You can also remember it using (30^\circ=\frac{\pi}{6}).

(\frac{11\pi}{6}\times\frac{180^\circ}{\pi}=330^\circ). After canceling (\pi), calculate (11\times30^\circ).

(\frac{5\pi}{4}\times\frac{180^\circ}{\pi}=225^\circ). Multiply (\frac{\pi}{4}=45^\circ) by (5).

(\frac{7\pi}{4}\times\frac{180^\circ}{\pi}=315^\circ). Recognize multiples of (45^\circ) quickly.

The direct answer is option A: 1 radian is approximately \(57.3^\circ\). A full revolution is \(360^\circ\), and it is also \(2\pi\) radians. Dividing both by \(2\pi\), one radian equals \(\frac{360^\circ}{2\pi}=\frac{180^\circ}{\pi}\). Using \(\pi\approx3.1416\), this is about 57.3 degrees. Option A matches. Option B, 45°, equals \(\frac{\pi}{4}\) radians, not 1 radian. Option C, 60°, is close but not the correct conversion; it is \(\frac{\pi}{3}\) radians. Option D, 90°, equals \(\frac{\pi}{2}\) radians. Thus A is the accurate approximate value. Remember the anchor conversions: \(\pi\) radians = 180°, so one radian is a little less than 60°.

The direct answer is option A: \(1^\circ\approx0.01745\) radians. Since 180° equals \(\pi\) radians, divide both sides by 180: \(1^\circ=\frac{\pi}{180}\) radians. With \(\pi\approx3.1416\), \(\frac{3.1416}{180}\approx0.01745\). Option A has exactly this value. Option B, 0.1745, is ten times too large and is close to 10° in radians. Option C, 1.745, is about 100° in radians, so it is far too large. Option D, 17.45, is also much too large and cannot represent a one-degree angle. A useful size check is that one degree is a very small angle, so its radian measure must be much less than 1. Remember: degree to radian means multiply by \(\frac{\pi}{180}\).

(35^\circ) lies between (0^\circ) and (90^\circ), so it is in the first quadrant. Identify quadrants using boundary angles.

(145^\circ) lies between (90^\circ) and (180^\circ), so it is in the second quadrant. First check the range of the angle.

(235^\circ) lies between (180^\circ) and (270^\circ), so it is in the third quadrant. Remember the range of the third quadrant.

(320^\circ) lies between (270^\circ) and (360^\circ), so it is in the fourth quadrant. The fourth quadrant comes after (270^\circ).

At (180^\circ), the terminal side lies on the negative (x)-axis. Angles on axes do not belong to any quadrant.

The governing concept is the position of a terminal side in standard position. An angle is measured anticlockwise from the positive x-axis. A quarter turn is 90°, a half turn is 180°, and three quarter turns are 270°. At 270°, the rotating ray points straight downward from the origin, which is the negative y-axis. Therefore option C is correct. The positive x-axis corresponds to 0° or 360°, the negative x-axis corresponds to 180°, and the positive y-axis corresponds to 90°. These are axis angles rather than interior points of quadrants, so it is better to identify the exact axis directly instead of applying only quadrant sign rules.

At (90^\circ), the terminal side lies on the positive (y)-axis. In standard position anticlockwise (90^\circ) goes upward.