## 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 $$0° \leqslant \theta \leqslant 2\pi \text{ rad}$$

Remark
 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}

Example

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

 (a) $$800^{\circ}$$ (b) $$\dfrac{19}{6}\pi \text{ rad}$$ Solution: (a) \begin{aligned} 800^{\circ} - 2(360^{\circ}) &= 80^{\circ}\\ 800^{\circ} &=2(360^{\circ}) +80^{\circ}\\ \end{aligned} $$\text{Thus, }800^{\circ} \text{ lies in Quadrant I}$$. (b) \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} $$\text{Thus, }\dfrac{19}{6}\pi \text{ rad}\text{ lies in Quadrant III}$$.