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.

Clockwise rotation gives a negative angle. In easy questions direction is the main clue.

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.

A quarter revolution is (360^\circ \div 4=90^\circ). Use division to solve such questions quickly.

The basic relation is (180^\circ=\pi) radians. Use it as the base for degree to radian conversion.

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.

(150^\circ \times \frac{\pi}{180^\circ}=\frac{5\pi}{6}). Simplify (150) and (180) by (30).

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

Multiplying \(\frac{\pi}{6}\) by \(180^\circ\) gives \(30^\circ\). The \(\pi\) cancels out.

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.

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

(\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}).