6.1 Positive Angles and Negative Angles
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 \(360^\circ\) (\(2\pi\text{ rad}\)).
Learning Objectives
- Represent positive angles and negative angles in a Cartesian plane.
- Convert angle measurements between degrees, minutes, and radians.
- Determine the position and quadrant of positive and negative angles, including angles exceeding \(360^\circ\) or \(2\pi\text{ rad}\).
Positive and Negative Rotations on the Cartesian Plane
See how the sign of an angle determines its rotation direction from the positive \(x\)-axis and how quadrants are numbered.
Rotation Directions and Quadrants
Key Ideas
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.
Short Explanation
Every angle starts at the positive \(x\)-axis. The sign tells you the direction of rotation.
1. Definition and Direction
Angles are measured starting from the positive \(x\)-axis about the origin \(O\). A positive angle rotates anticlockwise, while a negative angle rotates clockwise.
2. Quadrants of the Cartesian Plane
Quadrant I: \(0^\circ < \theta < 90^\circ\)
Quadrant II: \(90^\circ < \theta < 180^\circ\)
Quadrant III: \(180^\circ < \theta < 270^\circ\)
Quadrant IV: \(270^\circ < \theta < 360^\circ\)
3. Angle Unit Conversions
\(60' = 1^\circ\) and \(\pi\text{ rad} = 180^\circ\).
Degrees to radians: \(\theta^\circ \times \dfrac{\pi}{180^\circ}\)
Radians to degrees: \(\theta\text{ rad} \times \dfrac{180^\circ}{\pi}\)
4. Angles Greater than \(360^\circ\)
Subtract multiples of \(360^\circ\) or \(2\pi\text{ rad}\) until an equivalent angle between \(0^\circ\) and \(360^\circ\) is obtained.
| Quadrant | Degree range | Radian range |
|---|---|---|
| Quadrant I | \(0^\circ < \theta < 90^\circ\) | \(0 < \theta < \dfrac{\pi}{2}\text{ rad}\) |
| Quadrant II | \(90^\circ < \theta < 180^\circ\) | \(\dfrac{\pi}{2} < \theta < \pi\text{ rad}\) |
| Quadrant III | \(180^\circ < \theta < 270^\circ\) | \(\pi < \theta < \dfrac{3\pi}{2}\text{ rad}\) |
| Quadrant IV | \(270^\circ < \theta < 360^\circ\) | \(\dfrac{3\pi}{2} < \theta < 2\pi\text{ rad}\) |
Steps to Determine the Quadrant of an Angle Greater than \(360^\circ\)
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
\(\theta^\circ \times \dfrac{\pi}{180^\circ}\)Multiply by \(\dfrac{\pi}{180^\circ}\).
Radians to degrees
\(\theta\text{ rad} \times \dfrac{180^\circ}{\pi}\)Multiply by \(\dfrac{180^\circ}{\pi}\).
Try to Answer First
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?
Common Mistakes
- Measuring angles from the positive \(y\)-axis instead of the positive \(x\)-axis.
- Confusing rotation directions: treating clockwise as positive and anticlockwise as negative.
- Inverting the conversion factors between degrees and radians, for example multiplying by \(\dfrac{180^\circ}{\pi}\) when converting degrees to radians instead of multiplying by \(\dfrac{\pi}{180^\circ}\).
- Forgetting to account for negative signs when finding quadrants, for example assuming \(-135^\circ\) is in Quadrant II because \(135^\circ\) is in Quadrant II.
Concept Misunderstandings
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.
Summary
- Starting point: The positive \(x\)-axis.
- Direction: Anticlockwise = Positive angle; Clockwise = Negative angle.
- Quadrant limits: Quadrant I (\(0^\circ - 90^\circ\)), Quadrant II (\(90^\circ - 180^\circ\)), Quadrant III (\(180^\circ - 270^\circ\)), Quadrant IV (\(270^\circ - 360^\circ\)).
- Conversion formulas: \(\theta^\circ \times \dfrac{\pi}{180^\circ} = \text{radians}\); \(\theta\text{ rad} \times \dfrac{180^\circ}{\pi} = \text{degrees}\); \(60' = 1^\circ\).
- Cycles \(> 360^\circ\) or \(> 2\pi\text{ rad}\): Subtract multiples of \(360^\circ\) or \(2\pi\text{ rad}\) to find the corresponding terminal position.
Short Activity
Objective: Identify angle directions, quadrants and conversions.
A. Objective Quiz
1 Which of the following statements correctly defines a negative angle in trigonometry?
2 In which quadrant does the angle \(1\,000^\circ\) lie?
B. Fill in the Blanks
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 ______.
C. Matching / Drag and Drop
Drag each angle to the correct quadrant. On a phone, tap an angle first, then tap the matching box.
