Finding Theta Values With The Same Reference Angles

In trigonometry, the concept of reference angles is crucial for simplifying the evaluation of trigonometric functions for angles beyond the range of 0 to $\frac{\pi}{2}$ radians (0 to 90 degrees). Essentially, a reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. This acute angle provides a fundamental link between angles in different quadrants, allowing us to relate trigonometric values across the entire unit circle. To calculate a reference angle, you need to know in which quadrant the angle lies. This is because the way you determine the reference angle varies slightly depending on the quadrant. For instance, in the first quadrant (0 to $\frac{\pi}{2}$), the reference angle is simply the angle itself. However, in other quadrants, we need to perform some adjustments. In the second quadrant ($\frac{\pi}{2}$ to $\pi$), the reference angle is found by subtracting the given angle from $\pi$. In the third quadrant ($\pi$ to $\frac{3\pi}{2}$), the reference angle is calculated by subtracting $\pi$ from the given angle. Finally, in the fourth quadrant ($\frac{3\pi}{2}$ to $2\pi$), the reference angle is obtained by subtracting the given angle from $2\pi$. By mastering reference angles, we unlock a powerful tool that simplifies trigonometric calculations and deepens our understanding of angular relationships. Using reference angles, we can easily find the trigonometric functions of any angle, regardless of its size or quadrant location, by relating it back to an acute angle in the first quadrant. This concept also reinforces the periodic nature of trigonometric functions, as angles with the same reference angle will have trigonometric values that differ only in sign, depending on the quadrant.

When presented with a set of angles, identifying those that share the same reference angle involves understanding how angles are positioned within the four quadrants and how their terminal sides relate to the x-axis. The key to this identification process lies in calculating the reference angle for each given angle and then comparing these reference angles to see which ones match. Let’s consider a detailed breakdown of how to determine if angles have the same reference angle. First, you must ascertain the quadrant in which each angle's terminal side lies. This is vital because, as explained earlier, the calculation of the reference angle varies based on the quadrant. An angle in the first quadrant (0 to $\frac{\pi}{2}$) has a reference angle that is simply the angle itself. For an angle in the second quadrant ($\frac{\pi}{2}$ to $\pi$), you subtract the angle from $\pi$ to find its reference angle. In the third quadrant ($\pi$ to $\frac{3\pi}{2}$), you subtract $\pi$ from the angle to find the reference angle, and for angles in the fourth quadrant ($\frac{3\pi}{2}$ to $2\pi$), you subtract the angle from $2\pi$. Once you have calculated the reference angle for each given angle, you can compare them. Angles that yield the same reference angle are the ones you are looking for. This means that their terminal sides form the same acute angle with the x-axis, and their trigonometric values (sine, cosine, tangent, etc.) will be the same in magnitude, differing only in sign depending on the quadrant. For example, consider the angles $\frac{\pi}{3}$ (60 degrees) in the first quadrant and $\frac{2\pi}{3}$ (120 degrees) in the second quadrant. The reference angle for $\frac{\pi}{3}$ is simply $\frac{\pi}{3}$. The reference angle for $\frac{2\pi}{3}$ is calculated as $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$. Since both angles have the same reference angle of $\frac{\pi}{3}$, they are considered to have the same reference angle. This understanding helps streamline trigonometric calculations and enhances the comprehension of angular relationships within the unit circle.

Now, let's analyze the provided sets of angles to determine which sets contain angles that share the same reference angles. This involves applying the principles discussed earlier, quadrant by quadrant, and performing the necessary calculations to find the reference angle for each angle within the given sets. We will systematically go through each set, compute the reference angles, and identify the sets where all angles have the same reference angle. This process will demonstrate a practical application of the concept and clarify how to methodically approach such problems. Each set will be evaluated independently to ensure accuracy and clarity in our analysis. This step-by-step approach will not only help in solving this particular problem but also provide a general framework for handling similar questions involving reference angles.

Set 1: $\frac{\pi}{6}, \frac{\pi}{3}, \frac{5 \pi}{6}$

In this first set, we have three angles: $\frac{\pi}{6}$, $\frac{\pi}{3}$, and $\frac{5 \pi}{6}$. To determine if they share the same reference angle, we need to calculate the reference angle for each individually.

• For $\frac{\pi}{6}$, which lies in the first quadrant, the reference angle is the angle itself, so the reference angle is $\frac{\pi}{6}$.
• For $\frac{\pi}{3}$, also in the first quadrant, the reference angle is the angle itself, which is $\frac{\pi}{3}$.
• For $\frac{5 \pi}{6}$, located in the second quadrant, we calculate the reference angle by subtracting the angle from $\pi$: $\pi - \frac{5 \pi}{6} = \frac{\pi}{6}$.

Comparing the reference angles, we find that $\frac{\pi}{6}$ has a reference angle of $\frac{\pi}{6}$, $\frac{\pi}{3}$ has a reference angle of $\frac{\pi}{3}$, and $\frac{5 \pi}{6}$ has a reference angle of $\frac{\pi}{6}$. Therefore, this set does not have all angles with the same reference angle because $\frac{\pi}{3}$ has a different reference angle than the other two.

Set 2: $\frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3}$

Next, we examine the angles $\frac{\pi}{3}$, $\frac{5 \pi}{6}$, and $\frac{4 \pi}{3}$ to see if they share a common reference angle. We will proceed by finding the reference angle for each angle individually, just as we did in the previous set.

• For $\frac{\pi}{3}$, which is in the first quadrant, the reference angle is simply the angle itself, $\frac{\pi}{3}$.
• For $\frac{5 \pi}{6}$, situated in the second quadrant, we find the reference angle by subtracting it from $\pi$: $\pi - \frac{5 \pi}{6} = \frac{\pi}{6}$.
• For $\frac{4 \pi}{3}$, lying in the third quadrant, the reference angle is calculated by subtracting $\pi$ from the angle: $\frac{4 \pi}{3} - \pi = \frac{\pi}{3}$.

Upon comparing the reference angles, we observe that $\frac{\pi}{3}$ has a reference angle of $\frac{\pi}{3}$, $\frac{5 \pi}{6}$ has a reference angle of $\frac{\pi}{6}$, and $\frac{4 \pi}{3}$ has a reference angle of $\frac{\pi}{3}$. Consequently, this set also does not contain angles with the same reference angle, as $\frac{5 \pi}{6}$ has a distinct reference angle compared to the other two.

Set 3: $\frac{\pi}{2}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$

In this third set, we have the angles $\frac{\pi}{2}$, $\frac{5 \pi}{4}$, and $\frac{7 \pi}{4}$. Our goal remains the same: to determine the reference angle for each and identify if they are the same.

• For $\frac{\pi}{2}$, which lies on the positive y-axis, the concept of a reference angle is slightly different. It can be thought of as 0 since it forms a 90-degree angle with the x-axis. However, for consistency, we can consider its reference angle in the context of trigonometric functions, where $\frac{\pi}{2}$ is a quadrantal angle.
• For $\frac{5 \pi}{4}$, located in the third quadrant, the reference angle is found by subtracting $\pi$ from it: $\frac{5 \pi}{4} - \pi = \frac{\pi}{4}$.
• For $\frac{7 \pi}{4}$, which resides in the fourth quadrant, we calculate the reference angle by subtracting it from $2\pi$: $2\pi - \frac{7 \pi}{4} = \frac{\pi}{4}$.

When we compare these reference angles, we see that $\frac{\pi}{2}$ does not share the same reference angle as the other two. $\frac{5 \pi}{4}$ and $\frac{7 \pi}{4}$ both have a reference angle of $\frac{\pi}{4}$, but $\frac{\pi}{2}$ is different. Thus, this set does not have angles with the same reference angle across all three angles.

Set 4: $\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4}$

Finally, we analyze the set containing the angles $\frac{\pi}{4}$, $\frac{3 \pi}{4}$, and $\frac{7 \pi}{4}$. As with the previous sets, we will calculate the reference angle for each angle individually.

• For $\frac{\pi}{4}$, which is in the first quadrant, the reference angle is simply the angle itself: $\frac{\pi}{4}$.
• For $\frac{3 \pi}{4}$, located in the second quadrant, the reference angle is calculated by subtracting the angle from $\pi$: $\pi - \frac{3 \pi}{4} = \frac{\pi}{4}$.
• For $\frac{7 \pi}{4}$, situated in the fourth quadrant, we find the reference angle by subtracting the angle from $2\pi$: $2\pi - \frac{7 \pi}{4} = \frac{\pi}{4}$.

Comparing the calculated reference angles, we observe that $\frac{\pi}{4}$ has a reference angle of $\frac{\pi}{4}$, $\frac{3 \pi}{4}$ also has a reference angle of $\frac{\pi}{4}$, and $\frac{7 \pi}{4}$ likewise has a reference angle of $\frac{\pi}{4}$. Therefore, all angles in this set share the same reference angle.

In conclusion, after carefully analyzing each set of angles, we have identified that only the set {$\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4}$} contains angles that share the same reference angle. This determination was made by calculating the reference angle for each individual angle within each set and then comparing those reference angles. The angles in the set {$\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4}$} all have a reference angle of $\frac{\pi}{4}$, demonstrating a consistent relationship between their positions in different quadrants. This exercise underscores the importance of understanding reference angles in simplifying trigonometric calculations and appreciating the symmetries inherent in the unit circle.