Basics of Angles

Clockwise and Anticlockwise Rotations

Basic Angle, \(\alpha\)

The basic acute angle or the reference angle, \(\alpha\), is the acute angle that the given angle makes with the \(x\)-axis.

From O-Level to Additional Mathematics

In O-Level trigonometry you used sine, cosine and tangent of angles inside triangles. Additional Mathematics measures an angle as a rotation from the positive \(x\)-axis — anticlockwise for positive angles, clockwise for negative — so an angle can be any size. It also introduces the radian and three reciprocal ratios.

Degrees and radians

Conversion	Rule	Example
Degrees to radians
Radians to degrees

The quarter-turn markers are worth memorising: \(90^\circ=\dfrac{\pi}{2}\), \(180^\circ=\pi\), \(270^\circ=\dfrac{3\pi}{2}\), \(360^\circ=2\pi\).

Exam check

Match your calculator's angle mode to the question: degree questions carry the \(^\circ\) symbol, radian questions do not. A wrong mode gives a wrong answer even with correct working.

For an angle in any quadrant, the acute angle between the rotating arm and the \(x\)-axis is the basic angle (reference angle), \(\alpha\). The basic angle fixes the size of every ratio; the quadrant fixes the sign.

The bowtie shows equal basic angles measured from the \(x\)-axis — never from the \(y\)-axis. Reading the quadrants anticlockwise from the first, the letters A-S-T-C name what is positive there: All, Sine, Tangent, Cosine.

Basic angle and signs in each quadrant

Quadrant	Positive there	Angle \(\theta\) from basic angle \(\alpha\)	Sketch
I
II
III
IV

Negative angles and angles beyond \(360^\circ\)

Situation	Rule	Example
Negative angle (clockwise turn)
Beyond one revolution

Exam check

Bring the angle into \(0^\circ\) to \(360^\circ\) first, then read off the quadrant and the basic angle. The basic angle is always measured from the \(x\)-axis.

Video Solution:

Video solution locked

Solution:

Solution locked

Finding similar questions...

Need help? Join our JC Math tuition classes.

Learn more