Positive Angles and Negative Angles

6.1   Positive Angles and Negative Angles
  • Location of angles can be specified in terms of quadrants:
  • In trigonometry:
Positive Angles Negative Angles

Angles measured in the anti-clockwise direction from the positive \(x\)-axis


Angles measured in the clockwise direction from the positive \(x\)-axis

  • The position of an angle can be shown on a Cartesian plane
  • In general,

If \(\theta\) is an angle in a quadrant such that \(\theta \text{ > }360^{\circ}\),

then the position of \(\theta\) can be determined by substracting a multiple of \(360^{\circ}\) or \(2\pi \text{ rad}\) to obtain an angle that corresponds to

 \( 0° \leqslant \theta \leqslant 360^{\circ}\) or \( 0° \leqslant \theta \leqslant 2\pi \text{ rad}\)


The position of an angle can be specified by turning the angle in radian unit to degree unit:

  \(\begin{aligned} 60' &= 1^{\circ}\\\\ \theta^{\circ} &= \begin{pmatrix} \theta^{\circ} \times \dfrac{\pi}{180^{\circ}} \end{pmatrix}\\\\ \theta \text{ rad}&= \begin{pmatrix} \theta \text{ rad}\times \dfrac{180^{\circ}}{\pi} \end{pmatrix} \end{aligned}\)  

Determine the position of each of the following angles in the quadrants:

(a) \(800^{\circ}\)
(b) \(\dfrac{19}{6}\pi \text{ rad}\)

\(\begin{aligned} 800^{\circ} - 2(360^{\circ}) &= 80^{\circ}\\ 800^{\circ} &=2(360^{\circ}) +80^{\circ}\\ \end{aligned} \)

\(\text{Thus, }800^{\circ} \text{ lies in Quadrant I}\).


\(\begin{aligned} \dfrac{19}{6} \pi \text{ rad} - 2\pi \text{ rad} &= \dfrac{7}{6}\pi \text{ rad}\\ \dfrac{19}{6} \pi \text{ rad} &= 2\pi \text{ rad} + \dfrac{7}{6}\pi \text{ rad} \end{aligned} \)

\(\text{Thus, }\dfrac{19}{6}\pi \text{ rad}\text{ lies in Quadrant III}\).