Question

pi/3 is the reference angle for: A. 2pi/3 B. 15pi/3 C. 7pi/3 D. 19pi/3 **its CHOOSE ALL THAT APPLY!!!:) my answers: C. 7pi/3 and D. 19pi/3 are those right??

Answer 1

Answer:

The correct answers are A,C and D

Step-by-step explanation:

Given an angle θ, we call the reference angle for θ as '' θ ref ''.

Using the x-axis as our frame of reference, θ ref is always the smallest angle that you can make from the terminal side of an angle with the x-axis.

Let's note the quadrants of the IR2 as Q.

For an angle θ in QI, θ ref = θ

For an angle θ in QII, θ ref = π - θ

For an angle θ in QIII, θ ref = θ - π

For an angle θ in QIV, θ ref = 2π - θ

For A. $\frac{2\pi }{3}$

This angle is in QII ⇒ θ ref = π - θ

                                   θ ref = π - $\frac{2\pi }{3}$ = $\frac{\pi}{3}$

For B. $\frac{15\pi}{3}$

This angle is equal to $\frac{15\pi }{3}-4\pi  =\pi$

This angle is in QII ⇒ θ ref = π - θ

                                   θ ref = π - π = 0

For C. $\frac{7\pi }{3}$

This angle is equal to $\frac{7\pi }{3}-2\pi=\frac{\pi}{3}$

This angle is in QI ⇒ θ ref = θ = $\frac{\pi }{3}$

For D. $\frac{19\pi }{3}$

This angle is equal to $\frac{19\pi }{3}-6\pi=\frac{\pi }{3}$

This angle is in QI ⇒ θ ref = θ = $\frac{\pi }{3}$

The correct answers are A,C and D.

Notice that in the angles bigger than 2π we subtract multiples of 2π to know in which quadrants are the angles.

Answer 2

You can use the definition of reference angles to find out which angles are reference angles and which are not.

The angles for which the reference angle is pi/3 are given by

• Option A 2$\pi$/3
• Option C 7$\pi$/3
• Option D 19$\pi$/3

What is a reference angle of a given angle?

Think of reference angle as the minimum angle reaching from x axis to the terminal side of the given angle. Thus, if suppose the angle is 180 degrees, then it is overlapping on x axis, thus, the reference angle is 0.

If the angle is 135 degrees, we can reach it by shortcut from other side of x axis with only 45 degree walk. Thus, reference angle is 45 degrees.(see diagram attached below)

If its right angle, there is no choice, but only right angle to be as the reference angle.

How to find the reference angle of an angle?

We will have to check all options one by one

Option A: 2pi/3

Since we have $2\pi/3 + \pi/3 = \pi$

and $2\pi/3 = 120^\circ > 90^\circ$ thus, it is obtuse(in second quadrant). Thus, pi/3 is reference angle for this angle.

In reverse we could try finding reference angle of 2pi/3 and check if it is pi/3. We see that yes, since it is lying in second quadrant, thus, the negative side of x axis to that angle is reference angle whose measure, thus, will be pi - 2pi/3  = pi/3

Option B; 12pi/3

Since we have  $15\pi/3 = 5\pi = 2 \times 2 \pi + \pi = \pi$ = 180 degrees(lying on x axis) (remember 2pi + some angle = some angle as 2pi is full rotation)

The reference angle for this angle would be 0 degrees.

Thus, this option doesn't have pi/3 as its reference angle.

Option C: 7pi/3

Since we have $7\pi/3 = 2\pi + \pi/3 = \pi/3$

Since this angle is acute, its already minimally measured. Thus, its reference angle is it itself which is pi/3 (all angles here are measured in radians unless explicitly said degrees)

Option D: 19pi/3

Since we have $19\pi/3 = 3 \times 2\pi + \pi/3 = \pi/3$

This is same angle as given in previous option. Its reference angle is pi/3

Thus,

The angles for which the reference angle is pi/3 are given by

• Option A 2$\pi$/3
• Option C 7$\pi$/3
• Option D 19$\pi$/3

Learn more about reference angles here:

https://brainly.com/question/2697077