Angles

(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.

(1) radian (=\frac{180^\circ}{\pi}\approx57.3^\circ). For approximation take (\pi\approx3.14).

(1^\circ=\frac{\pi}{180}\approx0.01745) radian. A small degree angle has a small radian value.

(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.

At (270^\circ), the terminal side lies on the negative (y)-axis. Remember axial angles separately.

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

At (0^\circ), the terminal side stays with the initial side on the positive (x)-axis. A zero angle has no rotation.

(-90^\circ) is clockwise rotation so the terminal side lies on the negative (y)-axis. The negative sign shows direction.

There are 60 minutes in 1 degree. Thus, \(25^\circ=25\times60'=1500'\); adding the remaining \(30'\) gives \(1500'+30'=1530'\). \(1500'\) represents only 25 degrees and does not include the extra 30 minutes. Exam tip: To convert a mixed angle into minutes, multiply the degrees by 60 and then add the minutes.

(45'=\frac{45}{60}^\circ=0.75^\circ), so the total is (12.75^\circ). Divide minutes by (60).

Convert the decimal part 0.5° into minutes by multiplying it by 60: \(0.5\times 60'=30'\). Therefore, \(18.5^\circ=18^\circ 30'\). Option A has 15′, which equals \(0.25^\circ\), so it is not correct. Exam tip: To convert the decimal part of a degree into minutes, always multiply it by 60.