Smallest Positive X Where Sin(x+π) - Sin(x) = 1

Let's dive into the world of trigonometry and solve a fascinating problem! We're on a quest to find the smallest positive number, which we'll call x, that makes the equation sin(x + π) - sin(x) = 1 true. This isn't just about crunching numbers; it's about understanding the beautiful dance of sine waves and how they interact. So, grab your thinking caps, and let's get started!

Understanding the Sine Function and Trigonometric Identities

Before we jump into solving the equation, let's take a moment to appreciate the sine function. The sine function, often written as sin(x), is a fundamental concept in trigonometry. It describes the relationship between an angle in a right-angled triangle and the ratio of the length of the side opposite the angle to the length of the hypotenuse. But the sine function goes beyond triangles; it's a periodic function, meaning its values repeat in a regular pattern. This makes it incredibly useful for modeling cyclical phenomena like waves and oscillations. Think about sound waves, light waves, or even the motion of a pendulum – the sine function can help us understand them all.

Now, let's talk about trigonometric identities. These are like the secret ingredients in our mathematical toolkit. They're equations that are always true, no matter what value we plug in for the variable. One identity that's particularly useful for our problem is the sine addition formula: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). This formula allows us to break down the sine of a sum of angles into a more manageable form. It's like having a universal key that unlocks many trigonometric doors.

In our case, we have sin(x + π). Using the sine addition formula, we can rewrite this as sin(x)cos(π) + cos(x)sin(π). Remember that π (pi) is a special number in mathematics, approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter. The cosine of π is -1, and the sine of π is 0. So, our expression simplifies to sin(x)(-1) + cos(x)(0), which further simplifies to -sin(x). This is a crucial step in solving our problem because it transforms the original equation into something much easier to work with.

Solving the Equation sin(x+π) - sin(x) = 1

Now that we've armed ourselves with the sine addition formula and a good understanding of the sine function, let's tackle our equation: sin(x + π) - sin(x) = 1. As we discovered earlier, sin(x + π) can be simplified to -sin(x). So, we can rewrite our equation as -sin(x) - sin(x) = 1. Combining the terms, we get -2sin(x) = 1. This is starting to look much simpler, isn't it?

To isolate sin(x), we divide both sides of the equation by -2, giving us sin(x) = -1/2. Now, we're faced with a classic trigonometric problem: finding the angles whose sine is -1/2. Think about the unit circle, a circle with a radius of 1 centered at the origin of a coordinate plane. The sine of an angle corresponds to the y-coordinate of the point where the angle intersects the unit circle. So, we're looking for points on the unit circle where the y-coordinate is -1/2.

There are infinitely many angles that satisfy this condition because the sine function is periodic. However, we're only interested in the smallest positive value of x. If you visualize the unit circle or recall the sine values for common angles, you'll remember that sin(7π/6) = -1/2 and sin(11π/6) = -1/2. These angles are in the third and fourth quadrants, respectively, where the sine function is negative. The angle 7π/6 is equivalent to 210 degrees, and the angle 11π/6 is equivalent to 330 degrees.

Since we're looking for the smallest positive solution, we choose x = 7π/6. This is the first angle we encounter as we move counterclockwise from 0 along the unit circle where the sine is -1/2. Therefore, the smallest positive number x that satisfies the equation sin(x + π) - sin(x) = 1 is 7π/6. We've successfully navigated the trigonometric terrain and found our answer!

Verifying the Solution

It's always a good practice to verify our solution to make sure we haven't made any mistakes along the way. To do this, we'll plug x = 7π/6 back into our original equation: sin(x + π) - sin(x) = 1. Substituting 7π/6 for x, we get sin(7π/6 + π) - sin(7π/6) = 1.

First, let's simplify the argument of the first sine function: 7π/6 + π = 7π/6 + 6π/6 = 13π/6. So, we have sin(13π/6) - sin(7π/6) = 1. Now, we need to find the values of these sine functions. Remember that the sine function has a period of 2π, meaning sin(θ + 2π) = sin(θ) for any angle θ. We can use this property to simplify sin(13π/6).

13π/6 is more than 2π, so we can subtract 2π (which is 12π/6) to find an equivalent angle within the first period: 13π/6 - 12π/6 = π/6. Therefore, sin(13π/6) = sin(π/6), which is 1/2. We already know that sin(7π/6) = -1/2. So, our equation becomes 1/2 - (-1/2) = 1. This simplifies to 1/2 + 1/2 = 1, which is indeed true. Our solution is verified!

Alternative Approaches and Insights

While we've successfully solved the equation using the sine addition formula and the unit circle, it's always interesting to explore alternative approaches. Mathematics is like a vast landscape with many different paths leading to the same destination. Let's consider a different way to think about this problem.

We started with the equation sin(x + π) - sin(x) = 1. Notice that sin(x + π) represents a sine wave shifted by π radians (180 degrees) to the left. This shift has a specific effect on the sine wave: it flips it vertically. In other words, sin(x + π) is the negative of sin(x). We used the sine addition formula to prove this, but understanding this geometric interpretation can provide a more intuitive grasp of the problem.

Knowing that sin(x + π) = -sin(x), we can immediately simplify the equation to -sin(x) - sin(x) = 1, which leads to -2sin(x) = 1, just as before. This approach bypasses the need for the sine addition formula, offering a quicker route to the solution. It highlights the power of visualizing trigonometric functions and understanding their transformations.

Another interesting insight comes from considering the symmetry of the sine function. The sine function is symmetric about the origin, meaning sin(-x) = -sin(x). This property, along with the periodic nature of the sine function, allows us to find multiple solutions to trigonometric equations. In our case, we focused on finding the smallest positive solution, but there are infinitely many other solutions, both positive and negative.

Conclusion

We've successfully navigated the world of trigonometry to find the smallest positive number x that satisfies the equation sin(x + π) - sin(x) = 1. Our journey involved understanding the sine function, applying trigonometric identities, and visualizing the unit circle. We also explored alternative approaches and gained valuable insights into the nature of trigonometric functions. The answer, as we discovered, is x = 7π/6.

Trigonometry is a fascinating branch of mathematics with applications in various fields, from physics and engineering to music and art. By mastering its fundamental concepts and techniques, we can unlock a deeper understanding of the world around us. Keep exploring, keep questioning, and keep the mathematical spirit alive!

For further exploration of trigonometric functions and identities, consider visiting Khan Academy's trigonometry section.