One complete rotation of a ray in the anticlockwise direction results in an angle measuring of 360°.
By continuing the anticlockwise rotation, angles larger than 360° can be produced.
If we rotate in clockwise direction, negative angles are produced. Angles 57°, 417° and -303° have the same initial side and terminal side but with different amount of rotations, such angles are called coterminal angles.
Example 1 :
For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°.
(i) 395° (ii) 525° (iii) 1150° (iv) -270° (v) -450°
Solution :
(i) 395°
Write 395° in terms of 360°.
395° = 360° + 35°
So, the coterminal angle of 395° is 35^{◦}
(ii) 525°
Write 525° in terms of 360°.
525° = 360° + 165°
So, the coterminal angle of 525° is 165°.
(iii) 1150°
Write 1150° in terms of 360°.
1150° = 3(360°) + 70°
So, the coterminal angle of 1150° is 70°.
(iv) -270°
Write -270° in terms of 360°.
-270° = -360° + 90°
So, the coterminal angle of 270° is 90°.
(v) -450°
Write -450° in terms of 360°.
-450° = -360° - 90°
So, the coterminal angle of 450° is -90°.
Positive Angle Quadrant :
Angle lies between 0° and 90° -----> 1^{st} quadrant
Angle lies between 90° and 180° -----> 2^{nd} quadrant
Angle lies between 180° and 270° -----> 3^{rd} quadrant
Angle lies between 270° and 360° -----> 2^{th} quadrant
Negative Angle Quadrant :
Angle lies between 0° and -90° -----> 4^{th} quadrant
Angle lies between -90° and -180° -----> 3^{rd} quadrant
Angle lies between -180° and -270° -----> 2^{nd} quadrant
Angle lies between -270° and -360° -----> 1^{st} quadrant
Example 2 :
Identify the quadrant in which an angle of each given measure lies
(i) 25° (ii) 825° (iii) −55°
Solution :
(i) 25°
25° lies between 0° and 90°.
So, 25° lies in the first quadrant.
(ii) 825°
If the given angle measures more than 360°, then we have to divide the given angle by 360 and find the quadrant for the remaining angle.
When 825° is divided by 360°, the remainder is 105°.
105° lies between 90° and 180°.
So, 105° lies in the second quadrant.
(i) -55°
-55° lies between -90° and 0°.
So, -55° lies in the fourth quadrant.
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