Solving 2sinx - √2 = 0: A Step-by-Step Trigonometry Guide

Hey guys! Let's dive into the exciting world of trigonometry and tackle the equation $2 ext{sin} x - \sqrt{2} = 0$. This equation might seem intimidating at first, but don't worry, we'll break it down step by step. Our main goal here is to find all the solutions for $x$ within the intervals $[0, 2\pi)$ (radians) and $[0^{\circ}, 360^{\circ})$ (degrees). So, buckle up and let's get started!

Understanding Trigonometric Equations

Before we jump into solving this specific equation, let's make sure we're all on the same page about trigonometric equations. Basically, these equations involve trigonometric functions like sine, cosine, tangent, and their reciprocals (cosecant, secant, and cotangent). When solving these equations, we're looking for the angles that make the equation true. Think of it like a puzzle where the pieces are angles, and the picture we're trying to create is a balanced equation. The fundamental trigonometric functions, sine, cosine, and tangent, are the building blocks. Sine (sin) relates an angle to the ratio of the opposite side to the hypotenuse in a right triangle. Cosine (cos) relates an angle to the ratio of the adjacent side to the hypotenuse. Tangent (tan) relates an angle to the ratio of the opposite side to the adjacent side. Understanding these relationships is crucial for solving trigonometric equations. The unit circle is our best friend when it comes to visualizing these functions and their values at different angles. It’s a circle with a radius of 1, and the coordinates of points on the circle correspond to the cosine and sine of the angle formed with the positive x-axis. This visual aid helps us quickly identify angles with specific sine, cosine, or tangent values. Also, remember the periodic nature of trigonometric functions. Sine and cosine, for instance, repeat their values every $2\pi$ radians or $360^{\circ}$. This means there are infinitely many solutions to a trigonometric equation, but we're usually interested in the solutions within a specific interval, like the ones mentioned earlier. Before you even start solving, it’s helpful to have a good grasp of the unit circle and the values of trigonometric functions at common angles like $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$, and so on. This knowledge will significantly speed up the process and reduce the chances of making mistakes. When tackling trigonometric equations, always remember to consider the periodicity of the functions involved and the given interval for solutions. This will ensure you find all the solutions within the specified range. Trigonometric equations are more than just abstract mathematical problems; they have real-world applications in fields like physics, engineering, and navigation. Understanding how to solve them is a valuable skill that can help you analyze and model various phenomena. So, let’s keep practicing and exploring the fascinating world of trigonometry!

Solving 2sinx - √2 = 0 in Degrees

Let's tackle the equation $2 ext{sin} x - \sqrt{2} = 0$ within the interval $[0^{\circ}, 360^{\circ})$. Our first step is to isolate the $\text{sin} x$ term. We can do this by adding $\sqrt{2}$ to both sides of the equation, which gives us $2 ext{sin} x = \sqrt{2}$. Next, we divide both sides by 2 to get $\text{sin} x = \frac{\sqrt{2}}{2}$. Now, we need to think about what angles have a sine value of $\frac{\sqrt{2}}{2}$. This is where our knowledge of the unit circle comes in handy! If you recall the unit circle, you'll remember that $\text{sin} x$ corresponds to the y-coordinate of a point on the circle. We're looking for angles where the y-coordinate is $\frac{\sqrt{2}}{2}$. There are two angles within the interval $[0^{\circ}, 360^{\circ})$ that satisfy this condition. The first one is in the first quadrant, which is $45^{\circ}$. This is a common angle that you should recognize from the unit circle. The second angle is in the second quadrant. To find this angle, we can use the fact that $\text{sin} (180^{\circ} - x) = \text{sin} x$. So, the second angle is $180^{\circ} - 45^{\circ} = 135^{\circ}$. Therefore, the solutions in the interval $[0^{\circ}, 360^{\circ})$ are $x = 45^{\circ}$ and $x = 135^{\circ}$. Make sure to always double-check your solutions by plugging them back into the original equation. In this case, $2 ext{sin} 45^{\circ} - \sqrt{2} = 2(\frac{\sqrt{2}}{2}) - \sqrt{2} = 0$ and $2 ext{sin} 135^{\circ} - \sqrt{2} = 2(\frac{\sqrt{2}}{2}) - \sqrt{2} = 0$, so our solutions are correct! When dealing with trigonometric equations, it’s easy to miss solutions if you only focus on one quadrant. Always consider all possible quadrants where the trigonometric function can have the desired value. Understanding the symmetry of the unit circle and the properties of trigonometric functions is key to finding all solutions. Keep practicing, and you'll become a pro at solving these types of equations in no time!

Finding Solutions in Radians: 2sinx - √2 = 0

Now, let's find the solutions for the same equation, $2 ext{sin} x - \sqrt{2} = 0$, but this time in the interval $[0, 2\pi)$. We've already done the hard work of isolating $\text{sin} x$ and finding that $\text{sin} x = \frac{\sqrt{2}}{2}$. The key difference now is that we need to express our angles in radians instead of degrees. Just like before, we need to think about the unit circle and the angles where the y-coordinate (which represents the sine value) is $\frac{\sqrt{2}}{2}$. We already know that $45^{\circ}$ is one solution. To convert this to radians, we can use the conversion factor $\frac{\pi}{180^{\circ}}$. So, $45^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{4}$. Therefore, one solution is $x = \frac{\pi}{4}$. The other solution we found in degrees was $135^{\circ}$. Converting this to radians, we get $135^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{3\pi}{4}$. So, the second solution is $x = \frac{3\pi}{4}$. These are the only two solutions in the interval $[0, 2\pi)$. Remember that $2\pi$ radians is equivalent to $360^{\circ}$, so we're looking for angles within one full rotation of the unit circle. Always double-check your radian solutions to make sure they fall within the specified interval and that they satisfy the original equation. In this case, $\text{sin}(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\text{sin}(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}$, so our solutions are correct. Working with radians can sometimes feel a bit tricky at first, but the more you practice, the more comfortable you'll become with them. It's essential to understand the relationship between degrees and radians and to be able to convert between them quickly. Radians are the standard unit of angular measure in mathematics and physics, so mastering them is a valuable skill. Keep exploring the unit circle and practicing with trigonometric equations, and you'll become a radian expert in no time!

Final Answer

So, to wrap things up, we've successfully solved the equation $2 ext{sin} x - \sqrt{2} = 0$ in both degrees and radians. In the interval $[0^{\circ}, 360^{\circ})$, the solutions are $x = 45^{\circ}$ and $x = 135^{\circ}$. In the interval $[0, 2\pi)$, the solutions are $x = \frac{\pi}{4}$ and $x = \frac{3\pi}{4}$. Hopefully, this step-by-step guide has helped you understand the process of solving trigonometric equations a bit better. Remember, the key is to isolate the trigonometric function, think about the unit circle, and consider all possible solutions within the given interval. Don't be afraid to draw diagrams and use visual aids to help you visualize the problem. Practice makes perfect, so keep working on these types of equations, and you'll become a trigonometry whiz in no time! Always remember to double-check your work and make sure your answers make sense in the context of the problem. Trigonometry is a fascinating branch of mathematics with many real-world applications, so the effort you put into mastering it will definitely pay off. Keep exploring, keep learning, and have fun with it!

The solution(s) in $[0, 2\pi)$ is/are $x = \frac{\pi}{4}, \frac{3\pi}{4}$.