Co-terminal Angles

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Co-terminal angles

Lets be familiar with some basic concepts before moving on to co terminal and reference angles.

  • Unit circle is a circle of radius 1.That’s why it is called unit circle.
  • For positive angle move anticlockwise around the circle starting from positive x axis.
  • For negative angle move clockwise around the circle starting from positive x axis.
  • One complete revolution =360° or 2π radians.
  • Two complete revolutions = 720° or 4π  radians.

 

Co terminal angles :

The angles which share the same initial  and terminal sides , initial side being the positive x axis.

There are infinite number of co terminal angles that can be found for any given angle.

 

Coterminal angle for angle less than 360.

 To find positive co terminal angle of any given angle we  add 360 or multiple of 360.

To find  negative co terminal angle we  subtract 360 or multiple of 360.

If angle is in radians then we use 2π   in place of 360.

 

Example : Find a positive and negative co terminal angle for each of the following angles.

  1. 50°                           b)  -120°                                 c) 2π/3

Solution:

  1. 50

Positive co terminal : 50+360= 410°

Negative co terminal : 50- 360 = -310°

2.  -120

Positive co terminal angle : -120+360 =240°

Negative co terminal angle: -120-360= -480°

3. 2π/3

Positive co terminal angle:  \dpi{120} \frac{2\pi }{3}+2\pi = \frac{8\pi }{3}

Negative co terminal angle :  \dpi{120} \frac{2\pi }{3}-2\pi = \frac{-4\pi }{3}

 

 

Co terminal angle for angles greater than 360.

Keep on adding or subtracting 360 from the given angle until  we reach at an angle which is less than 360 .

We can also subtract or add any multiple of 360 depending on the given angle value.

 

Examples: Find a positive and negative co terminal angles for each of the  given angles.

a) 875                         b)-1120          

Solution:

a)875

To find positive co terminal angle we keep on subtracting 875 until we get an angle less than 360.

875-360 = 515

515-360 = 155

Or  we can also subtract suitable  multiple of 360

875-2(360) = 875-720 => 155

To find negative co terminal angle we subtract further 360 from 155 and get 155-360= -205°

 

b) -1120

To find negative co terminal angle keep on adding 360 to get an angle less than  360.

-1120 + 360= -760

-760+360 = -400

-400+360= -40

or we can add a suitable multiple of 360

-1120 +3(360)= -40

To find positive coterminal angle we further add 360 to -40

So positive coterminal angle is  -40+360 = 320

Two angles are coterminal if the difference between them is a multiple of 360° or 2π.

 

Example: Determine if the following pairs of angles are coterminal.

  • 30 and 390
  • 120 and 450
  • -260 and 100

Solution:

  • 30-390= -360 So this pair is coterminal.
  • 120 -450= -330 This pair is not coterminal.
  • -260-100 =-360 This pair is coterminal.

 

 

Practice problems:

 

  1. Find a coterminal angle for each of the following angles:
  • -330
  • 245
  • -7π/6
  • 3π/4

2. Find a positive and negative coterminal angle for each of the following.

  • 11π/3
  • -5π/6
  • 735
  • -1230

3. Determine if the following pairs of angles are coterminal.

  • -300 and -60
  • 450 and  90
  • -340 and 70

 

 

 

 

 

 

 

Answers: 1)

  • 30
  • -115
  • 5π/6
  • -5π/4

 

2)

1)  positive : 5π/3   negative : -π/3

2) positive : 7π/6     negative : -17π/6

3) positive: 15     negative : -345

4) positive :210   negative : -150

 

3)

1) No

2) Yes

3) No

 

 

 

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