Coterminal & Standard Angles
We define an angle as the union of two non-collinear rays that have a common starting point. The two rays are referred to as the arms of the angle and the common starting point is called its vertex.
We can interpret an angle by rotating a ray from one position to another. When we use this interpretation of an angle, the ray to begin with is called the initial side, and the final position of ray is called the terminal side as shown in the figure.
If we rotate the ray in anti-clockwise direction, the angle formed in this way is termed as positive angle. The angle formed by clockwise rotation of ray is termed as negative angle.
Angle in Standard Position
''An angle whose vertex lies at the origin of a rectangular coordinate system and whose initial side is directed along the positive direction of x-axis of that coordinate system is said to be in standard position.''
Angles shown in the figures below are in standard position.
Coterminal Angles
''Angles having the same initial and terminal sides are called coterminal angles.''
Let there be an angle `\angle`AOB with OA as the initial side and OB as the terminal side as shown in the figures below:
Note that all the three angles `\angle`AOB in the above figures have the same initial and terminal sides. Hence these are coterminal angles.
That is (`\theta`+2`\pi`), (`\theta`+4`\pi`) and in general `\theta`+2(k)`\pi` are all coterminal angles with `\theta` where k is an integer.
Note: If `\theta` is an angle measured in degrees, then `\theta \pm k(360^\circ)` , where k is an integer, is an angle coterminal with `\theta`.
If `\theta` is measured in radians, then `\theta \pm` 2(k)`\pi` is an angle coterminal with `\theta`.
Example 1:
Locate the following angles in standard position on a rectangular coordinate system.
- `135^\circ`
- `-640^\circ`
- `310^\circ`
Example 2:
Find two +ve and two -ve angles that are coterminal with `60^\circ`.
Solution: We are given that `\theta` = `60^\circ`
Two positive angles coterminal with `\theta` = `60^\circ` are:
`\theta` + `360^\circ` = `60^\circ` + `360^\circ` = `420^\circ` and `\theta` + 2(`360^\circ`) = `60^\circ` + `720^\circ` = `780^\circ`
Two negative angles coterminal with `\theta` = `60^\circ` are:
`\theta` - `360^\circ` = `60^\circ` - `360^\circ` = `- 300^\circ` and `\theta` - 2(`360^\circ`) = `60^\circ` - `720^\circ` = `-660^\circ`
Example 3:
Find one positive and one negative angle coterminal with `(2\pi)/3`
Solution: One positive angle coterminal with `(2\pi)/3` is:
`( 2\pi)/3` + 2`\pi` = `(2\pi+6\pi)/3` = `(8\pi)/3`
One negative angle coterminal with `(2\pi)/3` is:
`( 2\pi)/3- 2\pi` = `(2\pi - 6\pi)/3` = `( - 4\pi)/3`