Angles

(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 \(360^\circ\).

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

In this formula (s) represents arc length. Knowing the symbols helps in applying the formula correctly.

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.

(r=\frac{s}{\theta}=\frac{16}{2}=8) cm. Rearrange the formula according to the required quantity.

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.