Question

Which values for theta have the same reference angles

Answer 1

Answer: The correct option is 4.

Step-by-step explanation:

If the reference angle is
`\theta` then the angles
`\pi-\theta`,
`\pi+\theta` and
`2\pi-\theta` have same reference angle.

If the reference angle is
`(\pi)/(4)`, then the following angles have same reference angles.

`\pi-(\pi)/(4)=(3\pi)/(4)`

`\pi+(\pi)/(4)=(5\pi)/(4)`

`2\pi-(\pi)/(4)=(7\pi)/(4)`

Since the angles
`(\pi)/(4),(3\pi)/(4),(7\pi)/(4)` have same reference angle
`(\pi)/(4)`, therefore the correct option is 4.

In other options, the denominators terms are not equal, therefore the given angle can not be represent in the form of
`\pi-\theta`,
`\pi+\theta` and
`2\pi-\theta`, so the remaining options are incorrect.

Answer 2

(d) π/4, 3π/4, 7π/4

Step-by-step explanation:

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.

(a) denominators are 3 and 6; reference angles are different

(b) denominators are 3 and 6; reference angles are different

(c) denominators are 2 and 4; reference angles are different

(d) 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

_____

Additional comment

As with a lot of multiple-choice questions, application of a little "number sense" can eliminate a lot of the work of choosing the correct answer. Here, we need to know what a reference angle is and how it can be computed. We also need to know about addition and subtraction of fractions, and what is required for them to be equal.