Positive angle and negative angle

Positive Angles and Negative Angles

6.1

Positive Angles and Negative Angles

This image is an educational graphic titled 'Angles in Trigonometry.' It features two sections: one for positive angles and one for negative angles. - The left section is labeled 'Positive Angles' and states that these angles are measured in the anticlockwise direction from the positive x-axis. - The right section is labeled 'Negative Angles' and states that these angles are measured in the clockwise direction from the positive x-axis. The information is presented in a clear and simple format, with the title and sections highlighted for easy understanding.

Location of Angles in Cartesian Plane

Figure

Diagram depicting the four quadrants of a Cartesian plane, showcasing their respective angular locations.

Description

If \(\theta\) is an angle in a quadrant such that \(\theta \gt 360^\circ\), then the position of \(\theta\) can be determined by subtracting a multiple of \(360^\circ\) or \(2\pi\) rad to obtain an angle that corresponds to \(0^\circ \leq \theta \leq 360^\circ\) or \(0 \leq \theta \leq 2\pi\) rad.

Example

Question

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

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

Solution

(a)

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

\(80^\circ\) is the corresponding angle of \(800^\circ\).

Since \(80^\circ\) lies in Quadrant \(\text{I}\),

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

(b)

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

\(\dfrac{7}{6}\pi\) rad is the corresponding angle of \(\dfrac{19}{6}\pi\) rad.

Since \(\dfrac{7}{6}\pi\) rad lies in Quadrant \(\text{III}\),

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

Positive Angles and Negative Angles

6.1

Positive Angles and Negative Angles

Location of Angles in Cartesian Plane

Figure

Diagram depicting the four quadrants of a Cartesian plane, showcasing their respective angular locations.

Description

Example

Question

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

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

Solution

(a)

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

\(80^\circ\) is the corresponding angle of \(800^\circ\).

Since \(80^\circ\) lies in Quadrant \(\text{I}\),

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

(b)

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

\(\dfrac{7}{6}\pi\) rad is the corresponding angle of \(\dfrac{19}{6}\pi\) rad.

Since \(\dfrac{7}{6}\pi\) rad lies in Quadrant \(\text{III}\),

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

Positive Angles and Negative Angles

Positive Angles and Negative Angles

Chapter : Trigonometric Functions

Topic : Positive angle and negative angle

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