Solving 3 Cos X - 3 Sin X Cos X = 0 On [0°, 360°) A Comprehensive Guide
Hey guys! Let's dive into the exciting world of trigonometry and tackle the equation 3 cos x - 3 sin x cos x = 0 within the interval [0°, 360°). This problem is a fantastic example of how we can use trigonometric identities and algebraic techniques to find exact solutions. Whether you're a student brushing up on your skills or just a math enthusiast, this guide will walk you through each step in detail. We'll break down the problem, explore the concepts involved, and make sure you understand how to nail these types of questions. So, grab your pencils, and let's get started!
Understanding the Problem
Before we jump into solving, let’s make sure we fully grasp what the question is asking. We need to find all the angles x in the interval [0°, 360°) that satisfy the equation 3 cos x - 3 sin x cos x = 0. This means we’re looking for angles within one full rotation of the unit circle. The solutions should be exact values, which means we'll aim to express them using special angles (like 0°, 30°, 45°, 60°, 90°) and their corresponding trigonometric values.
The equation itself involves both cosine and sine functions, making it a bit trickier than a simple trigonometric equation. To solve it, we’ll need to use algebraic manipulation and possibly some trigonometric identities to simplify the equation into a form we can easily work with. Factoring will be our primary tool here, allowing us to break down the equation into smaller, more manageable parts. By setting each factor equal to zero, we can isolate the trigonometric functions and find the angles that satisfy the original equation. This process not only helps us find the solutions but also reinforces our understanding of how trigonometric functions behave within a given interval.
Step-by-Step Solution
Okay, let's roll up our sleeves and get into the nitty-gritty of solving this equation. Here's how we can break it down:
1. Factoring the Equation
The first thing we notice is that 3 cos x is a common factor in both terms of the equation. Factoring this out will simplify our equation significantly. So, let's factor 3 cos x from the equation 3 cos x - 3 sin x cos x = 0:
3 cos x (1 - sin x) = 0
2. Setting Factors to Zero
Now we have a product of two factors equaling zero. This means that at least one of the factors must be zero. We can set each factor equal to zero and solve for x:
3 cos x = 0 or 1 - sin x = 0
3. Solving for cos x = 0
Let's tackle the first equation: 3 cos x = 0. To isolate cos x, we divide both sides by 3:
cos x = 0
Now, we need to find the angles x in the interval [0°, 360°) where the cosine function equals zero. Recall that cosine corresponds to the x-coordinate on the unit circle. The x-coordinate is zero at 90° and 270°.
So, the solutions for cos x = 0 are:
- x = 90°
- x = 270°
4. Solving for sin x = 1
Next up is the second equation: 1 - sin x = 0. Let's isolate sin x by adding sin x to both sides:
sin x = 1
We need to find the angles x in the interval [0°, 360°) where the sine function equals one. Remember that sine corresponds to the y-coordinate on the unit circle. The y-coordinate is one at 90°.
So, the solution for sin x = 1 is:
- x = 90°
5. Combining the Solutions
Now, let's gather all the solutions we've found. From cos x = 0, we have x = 90° and x = 270°. From sin x = 1, we have x = 90°. Notice that x = 90° is a solution from both parts of our equation.
Therefore, the solutions to the equation 3 cos x - 3 sin x cos x = 0 in the interval [0°, 360°) are:
- x = 90°
- x = 270°
Alternative Approaches
While factoring is the most straightforward method for this equation, there are other approaches we could consider. Understanding different methods can give you a more comprehensive grasp of trigonometric equations. Let's explore a couple of alternative routes we could have taken.
1. Graphical Solution
One way to visualize the solutions is by graphing the equation y = 3 cos x - 3 sin x cos x and finding the points where the graph intersects the x-axis (i.e., where y = 0). You would plot the function over the interval [0°, 360°) and identify the x-values at these intersection points. This method is particularly useful for visualizing solutions and can be a great way to check your algebraic work. While it might not give you the exact values directly, it gives a visual confirmation of where the solutions lie.
2. Using Trigonometric Identities (Less Direct)
In some cases, you might think about using trigonometric identities to rewrite the equation. However, for this particular equation, using identities might complicate things more than necessary. For instance, you might be tempted to use the double-angle identity for sine (sin 2x = 2 sin x cos x), but that would not lead to an easier factorization in this specific case. Factoring remains the most efficient strategy here.
Understanding alternative methods is crucial because not all trigonometric equations can be solved in the same way. By expanding your toolkit of approaches, you'll be better equipped to tackle a wider range of problems. Whether it's through graphical analysis, trigonometric identities, or other algebraic techniques, the key is to identify the method that best simplifies the equation and leads you to the solutions.
Common Mistakes to Avoid
When solving trigonometric equations, it's easy to make a few common mistakes. Being aware of these pitfalls can help you avoid them and ensure you get the correct solutions. Let's take a look at some of the most frequent errors:
1. Forgetting to Factor
A very common mistake is trying to solve the equation without factoring first. In our case, if you didn't factor out 3 cos x, you'd have a much harder time isolating trigonometric functions. Factoring simplifies the equation and allows you to set each factor to zero, which is a crucial step in solving.
2. Dividing Both Sides by a Variable Expression
It might be tempting to divide both sides of the equation 3 cos x - 3 sin x cos x = 0 by cos x. However, this is a big no-no! When you divide by a variable expression (like cos x), you risk losing solutions. Why? Because you're essentially assuming that cos x is not zero. In our case, cos x = 0 gives us valid solutions (90° and 270°), so we would miss these if we divided by cos x. Always factor instead of dividing by variable expressions.
3. Not Considering the Interval
Another frequent mistake is finding solutions that are outside the specified interval. Our problem asked for solutions in the interval [0°, 360°). If we didn't pay attention to this, we might include solutions like -90° or 450°, which are not within the given range. Always double-check that your solutions fall within the interval specified in the problem.
4. Incorrectly Identifying Angles
Make sure you correctly identify the angles for which trigonometric functions have specific values. For example, confusing the angles where sin x = 1 or cos x = 0 can lead to incorrect solutions. A good way to avoid this is to have a strong understanding of the unit circle and the values of sine and cosine at special angles.
5. Missing Solutions
Sometimes, you might find one solution but miss others. For instance, in our equation, if you only solved for sin x = 1 and forgot about cos x = 0, you would miss two solutions. Always ensure you've considered all possible cases and solutions by thoroughly working through each factor.
By being mindful of these common mistakes, you can improve your accuracy and confidence in solving trigonometric equations. Always remember to factor, avoid dividing by variable expressions, stick to the given interval, and double-check your angles and solutions.
Real-World Applications
Trigonometric equations aren't just abstract math problems; they pop up in numerous real-world applications. Understanding how to solve them can be incredibly useful in various fields. Let's explore a few scenarios where these equations play a vital role:
1. Physics
In physics, trigonometric functions are essential for describing oscillatory motion, such as the motion of a pendulum or the vibration of a string. Equations involving sine and cosine are used to model these phenomena. For example, when analyzing simple harmonic motion, you might encounter equations similar to the one we solved, where you need to find the times (or angles) at which the displacement or velocity reaches a certain value. Solving these equations helps physicists predict and understand the behavior of oscillating systems.
2. Engineering
Engineers, particularly those in electrical and mechanical fields, frequently use trigonometric equations. In electrical engineering, alternating current (AC) circuits are described using sinusoidal functions. Finding the phase angles or the times when the current or voltage is at a specific level often involves solving trigonometric equations. Mechanical engineers use trigonometry to analyze forces and stresses in structures. Equations involving angles and trigonometric functions help in designing stable and efficient structures.
3. Navigation
Navigation systems, both traditional and modern, rely heavily on trigonometry. Calculating distances, bearings, and positions often requires solving trigonometric equations. For instance, in GPS (Global Positioning System) technology, satellites use trigonometric principles to determine the location of a receiver on Earth. Similarly, in nautical navigation, sailors use trigonometric functions to plot courses and determine their position based on landmarks or celestial bodies.
4. Computer Graphics and Game Development
Trigonometry is fundamental in computer graphics and game development. Creating realistic animations and 3D environments involves numerous calculations using trigonometric functions. Rotating objects, calculating trajectories, and creating perspective all require a solid understanding of trigonometric equations. Game developers use these equations to simulate realistic movements and interactions within the game world.
5. Acoustics
The study of sound and acoustics also relies on trigonometry. Sound waves can be modeled using sinusoidal functions, and analyzing the interference and superposition of sound waves often involves solving trigonometric equations. This is crucial in designing concert halls, audio equipment, and noise cancellation systems. Understanding the mathematical representation of sound helps engineers and scientists optimize audio experiences.
These examples illustrate just a fraction of the many real-world applications of trigonometric equations. From designing bridges to creating video games, the principles we've discussed are essential tools in a variety of fields. By mastering these concepts, you're not just solving math problems; you're building a foundation for understanding and contributing to the world around you.
Conclusion
Alright, guys, we've reached the end of our trigonometric journey! We successfully solved the equation 3 cos x - 3 sin x cos x = 0 within the interval [0°, 360°), finding the exact solutions x = 90° and x = 270°. We walked through the step-by-step process of factoring the equation, setting each factor to zero, and solving for the angles. We also explored alternative approaches, common mistakes to avoid, and real-world applications to give you a comprehensive understanding of the topic.
Remember, the key to mastering trigonometric equations is practice and a solid grasp of the fundamental concepts. Keep honing your skills by tackling different types of equations and exploring how these principles apply in various fields. Whether you're studying for an exam or just curious about the world around you, trigonometry is a powerful tool to have in your arsenal. So keep exploring, keep learning, and most importantly, keep having fun with math!
