Which values for $$ heta $$ have the same reference angles? $$ \frac{\pi}{6}, \frac{\pi}{3}, \frac{5 \pi}{6} $$ $$ \frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3} $$ $$ \frac{\pi}{2}, \frac{5 \pi}{4}, \frac{7 \pi}{4} $$ $$ \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4} $$

Question

Which values for $$ 	heta $$ have the same reference angles? $$ \frac{\pi}{6}, \frac{\pi}{3}, \frac{5 \pi}{6} $$ $$ \frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3} $$ $$ \frac{\pi}{2}, \frac{5 \pi}{4}, \frac{7 \pi}{4} $$ $$ \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4} $$

UpStudy AI · Accepted Answer

We need to find the set in which all angles have the same reference angle. The reference angle is defined as the acute angle between the terminal side of the angle and the horizontal axis.

1. For $$ \frac{\pi}{6} $$:
   - In Quadrant I, the reference angle is $$ \frac{\pi}{6} $$.

2. For $$ \frac{\pi}{3} $$:
   - In Quadrant I, the reference angle is $$ \frac{\pi}{3} $$.

3. For $$ \frac{5\pi}{6} $$:
   - In Quadrant II, the reference angle is 
     $$
     \pi - \frac{5\pi}{6} = \frac{\pi}{6}.
     $$
   - Here, $$ \frac{\pi}{6} $$ is different from $$ \frac{\pi}{3} $$.

Thus, the first set does not have all angles sharing the same reference angle.

Next, consider the set $$ \frac{\pi}{3}, \frac{5\pi}{6}, \frac{4\pi}{3} $$:

1. For $$ \frac{\pi}{3} $$ in Quadrant I:
   - The reference angle is $$ \frac{\pi}{3} $$.

2. For $$ \frac{5\pi}{6} $$ in Quadrant II:
   - The reference angle is 
     $$
     \pi - \frac{5\pi}{6} = \frac{\pi}{6}.
     $$
     
3. For $$ \frac{4\pi}{3} $$ in Quadrant III:
   - The reference angle is 
     $$
     \frac{4\pi}{3} - \pi = \frac{\pi}{3}.
     $$
     
Here, one angle has a reference angle of $$ \frac{\pi}{6} $$ while the others have $$ \frac{\pi}{3} $$. So this set does not work either.

Now, look at the set $$ \frac{\pi}{2}, \frac{5\pi}{4}, \frac{7\pi}{4} $$:

1. For $$ \frac{\pi}{2} $$:
   - This angle lies on the positive $$y$$-axis; the angle with the $$x$$-axis is $$ \frac{\pi}{2} $$. 
   - In general, reference angle is typically defined for angles not on the axes, and here $$ \frac{\pi}{2} $$ does not fit that typical definition.

2. For $$ \frac{5\pi}{4} $$ in Quadrant III:
   - The reference angle is 
     $$
     \frac{5\pi}{4} - \pi = \frac{\pi}{4}.
     $$

3. For $$ \frac{7\pi}{4} $$ in Quadrant IV:
   - The reference angle is 
     $$
     2\pi - \frac{7\pi}{4} = \frac{\pi}{4}.
     $$
     
Since $$ \frac{\pi}{2} $$ is markedly different, this set does not have all angles with the same reference angle.

Finally, consider the set $$ \frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4} $$:

1. For $$ \frac{\pi}{4} $$:
   - In Quadrant I, the reference angle is $$ \frac{\pi}{4} $$.

2. For $$ \frac{3\pi}{4} $$ in Quadrant II:
   - The reference angle is 
     $$
     \pi - \frac{3\pi}{4} = \frac{\pi}{4}.
     $$

3. For $$ \frac{7\pi}{4} $$ in Quadrant IV:
   - The reference angle is 
     $$
     2\pi - \frac{7\pi}{4} = \frac{\pi}{4}.
     $$

All three angles have the same reference angle of $$ \frac{\pi}{4} $$.

Thus, the set $$ \frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4} $$ is the correct answer.
The set $$ \frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4} $$ has all angles with the same reference angle of $$ \frac{\pi}{4} $$.

Which values for \( \theta \) have the same reference angles? \( \frac{\pi}{6}, \frac{\pi}{3}, \frac{5 \pi}{6} \) \( \frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3} \) \( \frac{\pi}{2}, \frac{5 \pi}{4}, \frac{7 \pi}{4} \) \( \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4} \)

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