Answer: the correct option is (d) π/4, 3π/4, 7π/4
The reference angle for an angle in standard position is the smallest angle between the terminal ray and the x-axis. Mathematically, it is the least of an angle θ and its supplement, where θ is the original angle modulo π (or 180°).
Observation
Here, all of the answer choices are given as positive fractions multiplying π. All of the values are less than 2π. When we compute their value modulo π, any fraction that results will have the same denominator as the fraction of the original angle. The same is true when we compute the angle's supplement: the denominator of the fraction will be unchanged
This means the reference angle will have the same denominator as the original angle.
Application
Since we want all of the angles to have the same reference angle, we can eliminate answer choices that cannot work, based on the denominators of the fractions.
denominators are 3 and 6; reference angles are different
denominators are 3 and 6; reference angles are different
denominators are 2 and 4; reference angles are different
denominators are all 4; reference angles may be the same
Check
If we actually compute the reference angles for answer choice 4, we find they are all π/4.
angle: π/4; supplement: 3π/4; minimum of these: π/4 = reference angle
angle: 3π//4; supplement: π/4; minimum of these: π/4 = reference angle
angle 7π/4; modulo π: 3π/4; supplement: π/4; minimum of these: π/4 =reference angle
Question: Which values for have the same reference angles?
The options that have the same reference angles are:
(a) π/6, 5π/6, 11π/
(b) π/3, 4π/3, 5π/3
(c) π/2, 3π/2, 5π/2
(d) π/4, 3π/4, 7π/4