Solving Sin²(x) + 5/14 = 6/7 Find X In [π/3, 4π/3]
Hey there, math enthusiasts! Today, we're diving into the exciting world of trigonometry to solve a problem that involves finding the values of x within a specific interval. Specifically, we aim to find the solutions for x in the equation sin²(x) + 5/14 = 6/7, where x lies between π/3 and 4π/3. Buckle up, because we're about to embark on a mathematical journey filled with trigonometric identities, algebraic manipulations, and a touch of critical thinking!
Understanding the Problem
Before we jump into the solution, let's take a moment to understand what we're dealing with. We have a trigonometric equation, sin²(x) + 5/14 = 6/7, which involves the sine function squared. Our mission is to find all the values of x that satisfy this equation, but with a twist! We're only interested in solutions that fall within the interval [π/3, 4π/3]. This constraint is crucial because trigonometric functions are periodic, meaning they repeat their values over and over again. Without this interval, we'd have infinitely many solutions. So, let's put on our thinking caps and get started!
Initial Equation Analysis
The initial equation we're tackling is sin²(x) + 5/14 = 6/7. This looks like a pretty standard trigonometric equation, but the presence of the sin²(x) term suggests that we might need to use some trigonometric identities or algebraic manipulations to isolate the sine function. Our first step should be to simplify the equation and see if we can get it into a more manageable form. Remember, guys, the key to solving any math problem is to break it down into smaller, more digestible steps. Let's get those algebraic skills warmed up!
To begin, we need to isolate the sin²(x) term. This can be achieved by subtracting 5/14 from both sides of the equation. This basic algebraic manipulation is crucial for simplifying the equation and moving closer to finding the values of x. This is a fundamental step that helps us clear the path for further trigonometric analysis.
Isolating the Trigonometric Function
To isolate sin²(x), we subtract 5/14 from both sides of the equation:
sin²(x) + 5/14 - 5/14 = 6/7 - 5/14
This simplifies to:
sin²(x) = 6/7 - 5/14
Now, we need to find a common denominator to subtract the fractions. The least common denominator for 7 and 14 is 14. So, we rewrite 6/7 as 12/14:
sin²(x) = 12/14 - 5/14
sin²(x) = 7/14
Further simplifying, we get:
sin²(x) = 1/2
Now we're getting somewhere! We've successfully isolated the sin²(x) term, and the equation looks much simpler. The next step is to take the square root of both sides, but we need to be careful to consider both positive and negative roots, as squaring either a positive or a negative number yields a positive result.
Taking the Square Root
Taking the square root of both sides of sin²(x) = 1/2 gives us:
sin(x) = ±√(1/2)
sin(x) = ±1/√2
To rationalize the denominator, we multiply the numerator and denominator by √2:
sin(x) = ±√2/2
So, we have two possible values for sin(x): √2/2 and -√2/2. This means we need to find all angles x within our interval [π/3, 4π/3] where the sine function takes on these values. This step is crucial as it branches our solution path into two possibilities, each requiring careful analysis within the given interval.
Finding the Angles
Now that we know sin(x) = √2/2 or sin(x) = -√2/2, we need to find the angles x that satisfy these conditions within the interval [π/3, 4π/3]. This involves recalling the unit circle and the values of sine at various standard angles. It’s like detective work, but with angles and trigonometric functions!
Analyzing sin(x) = √2/2
Let's start with sin(x) = √2/2. We know that sine corresponds to the y-coordinate on the unit circle. The angles where sin(x) = √2/2 are π/4 and 3π/4. However, we need to consider only the angles within our interval [π/3, 4π/3].
- The angle π/4 is not within our interval, as π/3 is approximately 1.047 radians and π/4 is approximately 0.785 radians. So, π/4 is too small.
- The angle 3π/4 is within our interval, as it is approximately 2.356 radians, which falls between π/3 (approximately 1.047 radians) and 4π/3 (approximately 4.189 radians). Thus, 3π/4 is a valid solution.
So, for sin(x) = √2/2, we have one solution within the given interval: x = 3π/4. This careful consideration of the interval is crucial to avoid extraneous solutions and ensures we only include values that fit the problem's constraints.
Analyzing sin(x) = -√2/2
Now, let's consider sin(x) = -√2/2. The angles where sin(x) = -√2/2 are 5π/4 and 7π/4. Again, we need to check which of these angles fall within our interval [π/3, 4π/3].
- The angle 5π/4 is within our interval, as it is approximately 3.927 radians, which falls between π/3 (approximately 1.047 radians) and 4π/3 (approximately 4.189 radians). So, 5π/4 is a valid solution.
- The angle 7π/4 is outside our interval, as it is approximately 5.498 radians, which is greater than 4π/3 (approximately 4.189 radians). So, 7π/4 is too large.
Thus, for sin(x) = -√2/2, we have one solution within the given interval: x = 5π/4. Remember, guys, always double-check that your solutions fall within the specified interval! It's a common mistake to overlook this, but it's crucial for getting the correct answer.
Final Solution
After carefully analyzing both cases, we have found two solutions for x within the interval [π/3, 4π/3]:
- x = 3π/4
- x = 5π/4
These are the angles that satisfy the equation sin²(x) + 5/14 = 6/7 within the given interval. We did it! We successfully navigated the trigonometric terrain and emerged victorious with the solutions we sought.
Therefore, the final answer is: x = 3π/4, 5π/4.
Graphical Verification
To further solidify our understanding and verify our solutions, let's consider a graphical approach. Visualizing the problem can often provide a deeper insight and help us confirm our algebraic findings. We'll plot the function y = sin²(x) + 5/14 and the horizontal line y = 6/7 on the same graph within the interval [π/3, 4π/3]. The points where the graph of the function intersects the line will represent the solutions to our equation.
Plotting the Functions
First, we plot the function y = sin²(x) + 5/14. This is a sinusoidal function with a minimum value of 5/14 and a maximum value of 1 + 5/14 = 19/14. The graph oscillates between these values with a period of π due to the sin²(x) term.
Next, we plot the horizontal line y = 6/7. This is a straight line parallel to the x-axis, representing a constant value.
Identifying Intersection Points
The points where the graph of y = sin²(x) + 5/14 intersects the line y = 6/7 are the solutions to our equation. By visually inspecting the graph within the interval [π/3, 4π/3], we can identify two intersection points. These points correspond to the x-values where the function and the line have the same y-value.
From the graph, we can estimate the x-coordinates of the intersection points. These estimates should align with our algebraic solutions, x = 3π/4 and x = 5π/4. This graphical verification provides a visual confirmation of our algebraic results, enhancing our confidence in the solutions.
Graph Interpretation
The graphical representation not only confirms our solutions but also provides a visual understanding of the problem. It shows how the sine function oscillates and how the solutions are the points where the function's value equals 6/7. This visual aid can be particularly helpful in understanding the periodic nature of trigonometric functions and how multiple solutions can exist within a given interval.
In summary, graphical verification is a powerful tool for confirming solutions to trigonometric equations. It provides a visual representation of the problem, allowing for a deeper understanding and increased confidence in the results.
Tips and Tricks for Trigonometric Equations
Solving trigonometric equations can be a bit like navigating a maze, but with the right strategies and a bit of practice, you can become a pro! Here are some tips and tricks to help you conquer those tricky equations:
- Simplify the Equation First: Before you start diving into trigonometric identities, always try to simplify the equation algebraically. This might involve combining like terms, factoring, or isolating trigonometric functions. Remember, guys, a simpler equation is a happier equation!
- Use Trigonometric Identities: Trigonometric identities are your best friends when solving these equations. Pythagorean identities (sin²(x) + cos²(x) = 1), double-angle formulas, and sum-to-product formulas can help you rewrite the equation in a more manageable form. Keep those identities handy!
- Isolate the Trigonometric Function: The goal is to isolate a single trigonometric function (sin(x), cos(x), tan(x), etc.) on one side of the equation. This will allow you to find the angles that satisfy the equation.
- Consider the Unit Circle: The unit circle is your visual guide to trigonometric values. Knowing the sine, cosine, and tangent of common angles (0, π/6, π/4, π/3, π/2, etc.) will make solving equations much easier. Visualize those angles!
- Find All Solutions within the Interval: Trigonometric functions are periodic, so they have infinitely many solutions. However, you're usually asked to find solutions within a specific interval. Make sure to find all the solutions within that interval and none outside of it. Interval awareness is key!
- Check Your Solutions: After you've found your solutions, plug them back into the original equation to make sure they work. This will help you catch any errors you might have made along the way. Always double-check!
- Graphical Verification: If possible, use a graphing calculator or software to graph the equation and visually verify your solutions. This can be a great way to confirm your answers and gain a deeper understanding of the problem.
- Practice, Practice, Practice: The more trigonometric equations you solve, the better you'll become at it. Don't be afraid to tackle challenging problems, and learn from your mistakes. Practice makes perfect, guys!
By following these tips and tricks, you'll be well on your way to mastering trigonometric equations. So, grab your pencils, warm up those brains, and start solving!
Conclusion
In conclusion, we've successfully solved the trigonometric equation sin²(x) + 5/14 = 6/7 within the interval [π/3, 4π/3]. We found that the solutions are x = 3π/4 and x = 5π/4. This journey involved algebraic manipulations, trigonometric identities, careful consideration of the unit circle, and a touch of graphical verification. What a ride, guys!
Solving trigonometric equations is a valuable skill in mathematics and has applications in various fields such as physics, engineering, and computer science. By mastering these techniques, you'll be well-equipped to tackle a wide range of problems. So, keep practicing and exploring the fascinating world of trigonometry!
Remember, the key to success in mathematics is understanding the underlying concepts, practicing regularly, and never being afraid to ask questions. Keep exploring, keep learning, and keep solving! You've got this!
