Determine the quadrant for each of the following angles. Then determine its reference angle (reference number).
(a) 462 degrees
(b) −290 degrees
(c) 6π/7 radians
(d) −16π/5 radians
(a) 462 degrees:
- 462 degrees is in the fourth quadrant.
- To find the reference angle, subtract 360 degrees from 462 degrees:
Reference angle = 462 degrees - 360 degrees = 102 degrees
(b) −290 degrees:
- -290 degrees is in the third quadrant.
- To find the reference angle, take the absolute value of -290 degrees and subtract it from 360 degrees:
Reference angle = 360 degrees - |-290 degrees| = 360 degrees - 290 degrees = 70 degrees
(c) 6π/7 radians:
- 6π/7 radians is in the second quadrant.
- To find the reference angle, subtract 6π/7 radians from π radians:
Reference angle = π radians - 6π/7 radians
= (7π/7) radians - (6π/7) radians
= π/7 radians
(d) −16π/5 radians:
- -16π/5 radians is in the third quadrant.
- To find the reference angle, take the absolute value of -16π/5 radians and subtract it from π radians:
Reference angle = π radians - |-16π/5 radians|
= π radians - (16π/5) radians
= π radians - (16/5)π radians
= (5π/5) radians - (16/5)π radians
= (5/5 - 16/5)π radians
= (-11/5)π radians