Quadrants & Quadrantal Angles

Quadrants


A rectangular coordinate system consists of x-axis and y-axis that divide a plane into four regions called quadrants. The point where the two axis intersect is called origin and is denoted by O.

Quadrants are usually denoted by roman letters I, II, III and IV as shown in the figure below:

quadrant
  • Angles between `0^\circ` and 9`0^\circ` are in the first quadrant.
  • Angles between `90^\circ` and `180^\circ` are in the second quadrant.
  • Angles between `180^\circ` and `270^\circ` are in the third quadrant.
  • Angles between `270^\circ` and `360^\circ` are in the fourth quadrant.

An angle in standard position is said to lie in a quadrant if its terminal side lies in that quadrant. In the figures below, `\apha` lies in the 1st quadrant, `\beta` lies in the 2nd quadrant, `\gamma` lies in the 3rd quadrant and `\theta` lies in the 4th quadrant.

first-quadrant second-quadrant third-quadrant fourth-quadrant

Quadrantal Angles


An angle in standard position is called a quadrantal angle if its terminal side lies on x-axis or y-axis. Quadrantal angles include `0^\circ`, `\pm90^\circ`, `\pm180^\circ`, `\pm270^\circ`, `\pm360^\circ` etc.

Some of these angles have been illustrated in the figures below. Note that the terminal side of these angles lies either on x-axis or on y-axis.

Quadrantal Angle Quadrantal Angle Quadrantal Angle

Example 1:

Indicate the quadrant of each of the following angles:

  1. `320^\circ`
  2. `- 135^\circ`
  3. `(7\pi )/ 4` radian
  4. `\pi` radian

Solution:

Since `320^\circ` lies between `270^\circ` and `360^\circ`, the angle lies in the 4th quadrant.

quadrant-angle

Since `-135^\circ` is coterminal with `-135^\circ` + `360^\circ` = `225^\circ` and `180^\circ` <`225^\circ` <`270^\circ`, therefore `225^\circ` lies in the 3rd quadrant and so does `-135^\circ`(Coterminal angles have same terminal side)

quadrant-angle225

Since `(7\pi)/4` radian = `315^\circ` , it lies in the 4th quadrant.

quadrant-angle315

Since `\pi` = `180^\circ` lies on x-axis, it is a quadrantal angle as shown below.

quadrant-angle180





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