In daily life, rotational movement can occur in two opposite directions: clockwise or anticlockwise. In trigonometry, angles are represented on a Cartesian plane using the positive \(x\)-axis as the starting reference line. Understanding whether a rotation is anticlockwise (positive angle) or clockwise (negative angle), as well as converting between degrees and radians, allows us to determine the exact quadrant where any angle lies, even when it exceeds one full rotation of \(360^\circ\) (\(2\pi\text{ rad}\)).
See how the sign of an angle determines its rotation direction from the positive \(x\)-axis and how quadrants are numbered.
Measured in the anticlockwise direction from the positive \(x\)-axis.
Measured in the clockwise direction from the positive \(x\)-axis.
The Cartesian plane is divided into four quadrants: I, II, III and IV, starting from the top-right and moving anticlockwise.
One complete rotation is \(360^\circ\) or \(2\pi\text{ rad}\). Angles greater than one full cycle continue into the next rotation.
Every angle starts at the positive \(x\)-axis. The sign tells you the direction of rotation.
Angles are measured starting from the positive \(x\)-axis about the origin \(O\). A positive angle rotates anticlockwise, while a negative angle rotates clockwise.
Quadrant I: \(0^\circ < \theta < 90^\circ\)Quadrant II: \(90^\circ < \theta < 180^\circ\)Quadrant III: \(180^\circ < \theta < 270^\circ\)Quadrant IV: \(270^\circ < \theta < 360^\circ\)
\(60' = 1^\circ\) and \(\pi\text{ rad} = 180^\circ\).Degrees to radians: \(\theta^\circ \times \dfrac{\pi}{180^\circ}\)Radians to degrees: \(\theta\text{ rad} \times \dfrac{180^\circ}{\pi}\)
Subtract multiples of \(360^\circ\) or \(2\pi\text{ rad}\) until an equivalent angle between \(0^\circ\) and \(360^\circ\) is obtained.
Identify the angle \(\theta\).
Subtract \(360^\circ\) or \(2\pi\text{ rad}\) repeatedly.
Obtain an equivalent angle between \(0^\circ\) and \(360^\circ\).
Compare with the quadrant ranges to determine its position.
Degrees to radians
Multiply by \(\dfrac{\pi}{180^\circ}\).
Radians to degrees
Multiply by \(\dfrac{180^\circ}{\pi}\).
Answer in your mind, then press "Check Answer".
What is the difference between a positive angle and a negative angle in a Cartesian plane?
Convert \(500^\circ\) into an equivalent angle between \(0^\circ\) and \(360^\circ\), and state its quadrant.
In which quadrant does the angle \(-135^\circ\) lie?
An angle greater than \(360^\circ\) cannot be represented on a Cartesian plane.
Angles greater than \(360^\circ\) are represented by completing one or more full rotations (\(360^\circ\) or \(2\pi\text{ rad}\)) and continuing the rotation into the corresponding quadrant.
A negative angle has a negative length or negative magnitude.
The negative sign merely denotes the direction of rotation (clockwise), not a negative physical size.
Objective: Identify angle directions, quadrants and conversions.
1 Which of the following statements correctly defines a negative angle in trigonometry?
2 In which quadrant does the angle \(1\,000^\circ\) lie?
3 Positive angles are angles measured in the ______ direction from the positive \(x\)-axis.
4 To convert an angle of \(\theta\) degrees into radians, multiply \(\theta^\circ\) by ______.
Drag each angle to the correct quadrant. On a phone, tap an angle first, then tap the matching box.
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