Question

Pi/6 is the reference angle for:

3pi/6

5pi/6

13pi/6

8pi/6

Answer 1

The reference angle for π/6 is π/6 itself, and among the given options, 5π/6 is the angle that has π/6 as its reference angle. The other angles have different reference angles.

Step-by-step explanation:

A reference angle is the smallest angle that a given angle makes with the x-axis on a standard position on a coordinate plane. π/6, or 30°, is the reference angle that we are comparing with the other angles provided in the question. The angles in radians are all multiples of π/6, and we can determine their quadrant and reference angle by how they relate to the standard position.

5π/6 is an angle that lies in the second quadrant and has a reference angle of π/6 because the second quadrant range is from π/2 to π; hence, subtracting this angle from π gives us the reference angle π - 5π/6 = π/6.

The angles 3π/6, 8π/6, and 13π/6 have reference angles different from π/6. To elaborate, 3π/6 simplifies to π/2, which is 90° and is its own reference angle since it lies on the y-axis. 8π/6 simplifies to 4π/3, which lies in the third quadrant, and the reference angle is 2π/3 - π/2 = π/6. 13π/6 is equivalent to 2π + π/6, a full rotation plus π/6, therefore the reference angle is π/6 directly since angles in standard position are typically considered modulo 2π.

Answer 2

`(5\pi)/(6), (13\pi)/(6)`

Step by step

Remember, a reference angle is a positive acute angle that represents an angle θ of any measurement.

a)

`(3\pi)/(6)=90^\circ`, then its reference angle is
`(\pi)/(2)`

b)

`(5\pi)/(6)=150^ \circ` is in the second cuadrant, then its reference angle is
`180^ \circ -150^(\circ)=30^(\circ)=(\pi)/(6)`

c)

`(13\pi)/(6)=390^\circ=360^\circ+30^\circ`, then it is in the first cuadrant and the reference angle is
`30^\circ =(\pi)/(6)`.

d)

`(8\pi)/(6)=240^\circ` and it is in the third cuadrant, then its reference angle is
`240^\circ -180^\circ=60^\circ=(\pi)/(3)`