Positive angle and negative angle

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)	\(\begin{aligned} \dfrac{19}{6} \pi \text{ rad} - 2\pi \text{ rad} &= \dfrac{7}{6}\pi \text{ rad}\\ \dfrac{19}{6} \pi \text{ rad} &= 2\pi \text{ rad} + \dfrac{7}{6}\pi \text{ rad} \end{aligned} \) \(\text{Thus, }\dfrac{19}{6}\pi \text{ rad}\text{ lies in Quadrant III}\).

Positive Angles and Negative Angles

Chapter : Trigonometric Functions

Topic : Positive angle and negative angle

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