Solving Trigonometric Equations: A Step-by-Step Guide
Hey math enthusiasts! Today, we're diving into the world of trigonometry to solve the equation 3 sin(θ/3) + 1 = 0 for θ within the interval [0°, 360°). Don't worry if this sounds a bit intimidating – we'll break it down into easy-to-follow steps, so you'll be solving these types of problems like a pro in no time. This guide is designed to be super clear and helpful, whether you're a student just starting out or someone who wants a refresher on trigonometric equations. Let's get started and make solving trig equations a breeze!
Isolating the Trigonometric Function
Alright, guys, the first thing we want to do when solving an equation like this is to isolate the trigonometric function. In our case, that's sin(θ/3). This means we need to get it all by itself on one side of the equation. It's like trying to get the main actor (the sine function) ready for their big scene! The equation we're working with is 3 sin(θ/3) + 1 = 0.
To isolate sin(θ/3), we'll need to do a couple of simple algebraic steps. First, subtract 1 from both sides of the equation. This gives us 3 sin(θ/3) = -1. See? We're making progress! Next, to completely isolate the sine function, divide both sides of the equation by 3. This yields sin(θ/3) = -1/3. Now, we've got the sine function all alone on one side, and we're ready to move on to the next step, which is all about finding the angle(s) whose sine is -1/3. Keep in mind, these are fundamental algebraic manipulations—the same rules you've been using since the beginning of your algebra journey. The goal here is to carefully undo the operations performed on the variable, working backward from the outermost operation to the innermost, like peeling layers off an onion. It's about unraveling the equation bit by bit until we can get to the heart of what we're trying to solve.
Remember, the key to algebra (and math in general) is to do the same thing to both sides of the equation to keep it balanced. In this context, we are working to isolate the trigonometric function; doing so helps us determine the angle(s) that satisfy the original equation. Isolating the function is the initial but crucial step, enabling us to use our knowledge of trigonometric functions and their properties to find the unknown angles. We're looking for angles, measured in degrees, that when their sine value is divided by 3, we get -1/3. This step is not merely procedural; it's about understanding how trigonometric functions behave and how to manipulate them to find solutions. Remember, solving trigonometric equations involves finding the values of the variable (in this case, theta) that satisfy the given equation. The process involves isolating the trigonometric function and using the properties of these functions to identify the angles that meet the equation's conditions.
Finding the Reference Angle
Okay, now that we have sin(θ/3) = -1/3, we need to figure out the angles that have a sine value of -1/3. Since the sine function is negative, we know that the angles we're looking for are in the third and fourth quadrants of the unit circle. Before we worry about those quadrants, let's find the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. We can find this by taking the inverse sine (also known as arcsin) of the absolute value of -1/3, which is 1/3.
So, using a calculator, we find that arcsin(1/3) ≈ 19.47°. This is our reference angle, often denoted as θ_ref. It's important to remember that this angle is in the first quadrant, where all trigonometric functions are positive. Knowing the reference angle helps us identify the angles in the third and fourth quadrants that will give us the desired sine value. In the third quadrant, the angle is 180° plus the reference angle, and in the fourth quadrant, the angle is 360° minus the reference angle.
Think of the reference angle as a compass to guide us to the correct solutions. The process of determining the reference angle and then using it to find the solutions in the appropriate quadrants is a cornerstone of solving trigonometric equations. We're essentially leveraging our understanding of the unit circle and the properties of sine to find the angles that satisfy our equation. The reference angle provides a foundation for our calculations, allowing us to navigate through the different quadrants of the unit circle and pinpoint the solutions.
The use of the inverse sine function is crucial. It reverses the sine function, giving us an angle from a sine value. By focusing on the absolute value, we ensure that the reference angle is always acute, making it easier to locate the correct angles in the third and fourth quadrants. Therefore, by finding the reference angle, we bridge the gap from the simple trigonometric ratio to the complete solution set within the specified interval. This step is crucial in accurately solving the trigonometric equation, requiring both the reference angle's computation and understanding of trigonometric relationships within different quadrants of the unit circle.
Determining the Angles in the Correct Quadrants
Alright, we've got our reference angle, and we know that the sine function is negative in the third and fourth quadrants. Now it's time to calculate the actual angles θ/3 that satisfy the equation. For the third quadrant, we add the reference angle to 180°: 180° + 19.47° = 199.47°. For the fourth quadrant, we subtract the reference angle from 360°: 360° - 19.47° = 340.53°. Remember, these are the values for θ/3, not θ. Keep in mind that we're looking for values of θ that, when divided by 3, result in a sine of -1/3. These angles represent the positions on the unit circle where the sine value is -1/3, and understanding these angles is a key part of the solution process. Now that we have θ/3, let's solve for θ to complete our journey.
The calculation of angles in the third and fourth quadrants is an important step in solving trigonometric equations, as it directly determines the specific angles within the unit circle. When we deal with trigonometric functions, the values of the function change depending on which quadrant the angle lies in. Because the sine function is negative, the solutions can only exist in the third and fourth quadrants. Hence, determining these specific angles requires understanding how the reference angle is adjusted within these quadrants. The reference angle is added to 180° in the third quadrant, and it is subtracted from 360° in the fourth quadrant. It is critical to grasp these manipulations, as they establish the foundational knowledge of angle measurement and the relationships between angles in various quadrants.
By accurately calculating the angles in the correct quadrants, we ensure that our solutions are precise and correctly correspond to the original trigonometric equation. This step is an example of how mathematical principles come together to solve problems. The solutions in these quadrants are the core of our trigonometric equation's answer, reflecting the positions on the unit circle where the sine function gives us the required negative value. It's a demonstration of how knowledge of the unit circle, combined with reference angles, allows us to pinpoint the exact solutions. The calculations involved highlight the essential need to comprehend trigonometric functions and their properties, enabling us to solve equations with accuracy.
Solving for θ
We have two values for θ/3: 199.47° and 340.53°. To find θ, we need to multiply both of these values by 3. So,
θ = 199.47° * 3 ≈ 598.41°θ = 340.53° * 3 ≈ 1021.59°
However, we need to remember the original interval: [0°, 360°). Both of these values for θ are outside of that interval. This means that we need to adjust our solutions to fit within the given interval. One of the key things that we should know is the periodicity of a trigonometric function. So, we can use the periodicity of the function to find the appropriate values. Since we know the period is 360°, we can subtract 360° from any angle to get a coterminal angle. Therefore, we can subtract 360° from both of the answers and find them within our original interval.
Let's subtract 360° from 598.41°:
598.41° - 360° ≈ 238.41°
Now, let's subtract 360° from 1021.59°:
1021.59° - 2 * 360° ≈ 301.59°
So, we found our solutions that are within the original given interval. We can summarize all the answers and write them in the simplest form. These are the final answers.
Conclusion
And there you have it! The solutions for θ in the interval [0°, 360°) for the equation 3 sin(θ/3) + 1 = 0 are approximately 238.41° and 301.59°. We've successfully navigated this trigonometric equation, breaking it down step by step to ensure clarity and understanding. Congratulations! You've taken another step toward mastering trigonometry. Keep practicing, and you'll get more comfortable with these types of problems. If you enjoyed this guide, and if you have any questions, feel free to drop them in the comments below!
Summary of Steps:
- Isolate the trigonometric function:
sin(θ/3) = -1/3 - Find the reference angle:
arcsin(1/3) ≈ 19.47° - Determine angles in the correct quadrants:
180° + 19.47° = 199.47°and360° - 19.47° = 340.53° - Solve for θ:
θ ≈ 598.41°andθ ≈ 1021.59° - Adjust solutions to fit the interval
[0°, 360°):θ ≈ 238.41°andθ ≈ 301.59°
Final Answer:
The solutions are approximately 238.41° and 301.59°.
