Ordering Sine Values: A Step-by-Step Guide

Hey guys! Today, we're diving into a super interesting problem: how to arrange sine values in ascending order. Specifically, we'll be tackling the sines of various angles, namely $\sin(\frac{\pi}{14})$, $\sin(\frac{\pi}{10})$, $\sin(\frac{\pi}{3})$, and $\sin(\frac{7\pi}{12})$. This might seem daunting at first, but trust me, we'll break it down into manageable steps. So, grab your thinking caps, and let's get started!

Understanding Sine Values

First things first, let's refresh our understanding of sine values. Remember the unit circle? The sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This is crucial because it immediately tells us that as the angle increases from 0 to $\frac{\pi}{2}$ (90 degrees), the sine value increases from 0 to 1. Sine is a fundamental trigonometric function, and grasping its behavior is key to solving problems like these. We need to understand how the sine function behaves within different quadrants of the unit circle to accurately compare the sine values of the given angles. The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants. However, in this specific problem, all the angles are within the first and second quadrants, so we primarily need to focus on how the sine function behaves in these quadrants. Understanding the sine function's periodic nature and its symmetry properties can further simplify the comparison process. For instance, $\sin(x) = \sin(\pi - x)$, which could be useful if we had angles in both the first and second quadrants. However, in this case, all angles are less than $\frac{7\pi}{12}$, which is less than $\pi$, so we're primarily dealing with positive sine values. The relationship between sine and cosine, specifically $\sin^2(x) + \cos^2(x) = 1$, is also a foundational trigonometric identity that helps us understand the interrelation between sine and cosine functions, though it might not be directly applicable in this particular ordering problem. Instead, we're primarily focused on the monotonic nature of the sine function within the first quadrant. This understanding will provide us with a strong foundation for accurately ordering the given sine values.

Converting to Degrees (Optional but Helpful)

Sometimes, working with radians can be a bit abstract. If you're more comfortable with degrees, we can convert our angles from radians to degrees. Remember, $\pi$ radians is equal to 180 degrees. So:

• $\frac{\pi}{14}$ radians = $\frac{180}{14}$ degrees ≈ 12.86 degrees
• $\frac{\pi}{10}$ radians = $\frac{180}{10}$ degrees = 18 degrees
• $\frac{\pi}{3}$ radians = $\frac{180}{3}$ degrees = 60 degrees
• $\frac{7\pi}{12}$ radians = $\frac{7 * 180}{12}$ degrees = 105 degrees

Converting to degrees isn't strictly necessary, but it can provide a more intuitive feel for the angles we're dealing with. For instance, seeing 105 degrees immediately gives us a sense of where the angle lies on the unit circle. This step is more about personal preference; if you're comfortable working with radians, you can skip it. However, for many, degrees are more familiar and easier to visualize. This conversion allows us to use our understanding of the unit circle in degrees to estimate the relative sine values of each angle. By comparing the angles in degrees, we can easily see which angles are smaller and larger, which helps in ordering their sine values since sine is an increasing function in the first quadrant and has a well-defined behavior in the second quadrant. Moreover, expressing the angles in degrees can help us leverage known sine values for special angles, such as 30, 45, 60, and 90 degrees, as benchmarks for comparison. This conversion process ultimately simplifies the ordering process by providing a more tangible representation of the angles in question.

Analyzing the Sine Values

Now, let's analyze the sine values themselves. We know that the sine function increases from 0 to 1 as the angle increases from 0 to $\frac{\pi}{2}$ (90 degrees). After $\frac{\pi}{2}$, it decreases back to 0 as the angle goes from $\frac{\pi}{2}$ to $\pi$ (180 degrees). All our angles are between 0 and $\pi$, so we're in good shape!

Since sine is increasing in the first quadrant (0 to 90 degrees), we can say that if angle A < angle B (and both are less than 90 degrees), then sin(A) < sin(B). This is a key principle for ordering our first three sine values. For angles greater than 90 degrees, we need to consider that sine starts decreasing. However, $\frac{7\pi}{12}$ (105 degrees) is the only angle greater than 90 degrees, so we'll address it separately. The critical part here is recognizing the monotonic behavior of the sine function. This means that within a certain range (specifically, 0 to $\frac{\pi}{2}$), as the angle increases, the sine value also increases. Understanding this allows us to directly compare sine values based on the corresponding angles, making the ordering process much simpler. For angles beyond $\frac{\pi}{2}$, we must account for the sine function's decrease, but in this case, we only have one such angle, which simplifies our comparison. By focusing on the increasing nature of sine in the first quadrant, we can establish a clear order for most of the given angles. This principle of monotonic increase is a fundamental tool for comparing trigonometric functions and forms the backbone of our analysis.

Ordering the Values

Okay, let's put it all together. We have the angles in radians: $\frac{\pi}{14}$, $\frac{\pi}{10}$, $\frac{\pi}{3}$, and $\frac{7\pi}{12}$. We also have their approximate degree equivalents: 12.86, 18, 60, and 105 degrees.

1. The smallest angle is $\frac{\pi}{14}$ (12.86 degrees), so $\sin(\frac{\pi}{14})$ will be the smallest sine value.
2. Next is $\frac{\pi}{10}$ (18 degrees), so $\sin(\frac{\pi}{10})$ is the second smallest.
3. Then comes $\frac{\pi}{3}$ (60 degrees), making $\sin(\frac{\pi}{3})$ the third smallest.
4. Finally, we have $\frac{7\pi}{12}$ (105 degrees). Since sine decreases after 90 degrees, we need to be a bit careful. However, sine is positive in the second quadrant, and 105 degrees is not that far past 90 degrees, so it will still be a relatively large value. In fact, because $\sin(\frac{\pi}{3})$ = $\sin(60°)$ = $\frac{\sqrt{3}}{2}$ ≈ 0.866, and $\sin(\frac{7\pi}{12})$ = $\sin(105°)$ = $\sin(75°)$ which is bigger since sine increases till 90 degrees. $\sin(75°)$ = $\sin(45° + 30°) = \sin45°\cos30° + \cos45°\sin30° $ so it is approximately 0.966. So $\sin(\frac{7\pi}{12})$ will be the largest.

Therefore, the sine values in ascending order are: $\sin(\frac{\pi}{14})$, $\sin(\frac{\pi}{10})$, $\sin(\frac{\pi}{3})$, $\sin(\frac{7\pi}{12})$.

To recap, we compared the angles and their positions relative to each other within the unit circle, remembering that the sine function increases from 0 to $\frac{\pi}{2}$ and then decreases. We used the degree equivalents as a helpful visual aid, and by carefully considering each angle, we confidently arranged the sine values in ascending order. The comparison process hinged on understanding the behavior of the sine function and its monotonic properties within specific intervals. By focusing on the increasing nature of the sine function in the first quadrant and accounting for its decrease in the second quadrant, we were able to accurately place each sine value in the correct order. This approach showcases the importance of not just knowing the definitions of trigonometric functions but also understanding their behavior and how they change with respect to angle variations. By thinking through the steps logically and applying our knowledge of sine, we were able to arrive at the correct ascending order with a clear understanding of the underlying trigonometric principles.

Final Answer

So, the final answer, with the sine values arranged in ascending order, is:

$\sin(\frac{\pi}{14}) < \sin(\frac{\pi}{10}) < \sin(\frac{\pi}{3}) < \sin(\frac{7\pi}{12})$

Awesome job, guys! We successfully navigated the world of sine values and arranged them in ascending order. Remember, understanding the behavior of trigonometric functions and using the unit circle as a visual aid can make these types of problems much easier. Keep practicing, and you'll become a sine-ordering pro in no time! This skill is essential for anyone delving deeper into trigonometry and calculus. Mastering trigonometric function comparison is a significant step in mathematical fluency. We've explored not just the mechanical steps of comparing the sine values but also the conceptual underpinnings that make this ordering possible. Understanding the monotonic nature of sine, the unit circle representation, and the radian-degree conversion process empowers you to tackle more complex trigonometric problems with confidence. The ability to visualize and reason about trigonometric functions is a key asset in mathematical problem-solving and expands your understanding of mathematical relationships. So, keep practicing and pushing your mathematical boundaries, and you'll find that these skills will serve you well in your mathematical journey.