Solving Trigonometric Equation: Finding X In [π/4, 5π/4]
Hey guys! Today, we're diving into a fun trigonometric problem where we need to find the values of x that satisfy a given equation within a specific interval. This is a classic problem that combines trigonometric identities and algebraic manipulation, so buckle up and let's get started!
Understanding the Problem
So, the problem we're tackling is this: Find all values of x that satisfy the equation 12cos²x - 3 = sec²x - tan²x + 5, given that x lies in the interval [π/4, 5π/4]. This means we're looking for solutions within a specific range on the unit circle. To effectively solve this, we’ll leverage key trigonometric identities, simplify the equation, and then find the angles that fit our criteria. Mastering trigonometric equations is a fundamental skill in mathematics, especially in calculus and physics, making this a valuable exercise. Our goal is to rewrite the equation in a simpler form, ideally involving a single trigonometric function, which will allow us to isolate the variable x and find its possible values. Remember, trigonometric equations often have multiple solutions due to the periodic nature of trigonometric functions, so it’s crucial to consider all possibilities within the given interval.
Breaking Down the Given Equation
At first glance, the equation 12cos²x - 3 = sec²x - tan²x + 5 might seem a bit intimidating, but don't worry, we'll break it down step by step. The left side involves a cosine squared term, while the right side involves secant and tangent squared terms. This is a classic setup where trigonometric identities can come to our rescue. The key here is to recall the fundamental trigonometric identity: sec²x = 1 + tan²x. This identity is derived from the Pythagorean identity sin²x + cos²x = 1, by dividing each term by cos²x. By applying this identity, we can simplify the right side of our equation and make it much more manageable. We also need to keep in mind that we are looking for solutions within the interval [π/4, 5π/4], which covers the first, second, and third quadrants of the unit circle. This means we need to consider where the cosine function (which appears in our equation) is positive and negative within this range. Remember, cosine is positive in the first quadrant and negative in the second and third quadrants. By understanding these basic concepts and identities, we're well-equipped to transform and solve this trigonometric equation. Let's dive deeper into the simplification process to see how this identity will help us.
Applying Trigonometric Identities
Alright, let's get our hands dirty with some trigonometric magic! As we discussed, the star of the show here is the identity sec²x = 1 + tan²x. This identity will help us simplify the right side of our equation: 12cos²x - 3 = sec²x - tan²x + 5. Now, let's substitute sec²x with 1 + tan²x in our equation. This gives us: 12cos²x - 3 = (1 + tan²x) - tan²x + 5. Notice anything cool? The tan²x terms cancel each other out! This leaves us with a much simpler equation: 12cos²x - 3 = 1 + 5, which further simplifies to 12cos²x - 3 = 6. See how a seemingly complex equation can be tamed with the right tools? Now that we've simplified the equation, we're in a much better position to isolate the cosine term and solve for x. This step highlights the power of trigonometric identities in simplifying expressions and making problems more approachable. By recognizing and applying these identities, we can transform complex equations into manageable forms, paving the way for finding solutions. Next, we'll isolate the cosine squared term and see what values we get.
Solving the Simplified Equation
Now that we've simplified the equation to 12cos²x - 3 = 6, let's isolate the cos²x term. This involves some basic algebraic manipulation. First, we add 3 to both sides of the equation: 12cos²x = 9. Then, we divide both sides by 12: cos²x = 9/12. We can simplify the fraction 9/12 to 3/4, so we have: cos²x = 3/4. Awesome! We're getting closer to finding x. But remember, we have cos²x, not just cos x. To find cos x, we need to take the square root of both sides. When we do this, we need to consider both the positive and negative square roots, since squaring either a positive or a negative number will give us a positive result. So, we have: cos x = ±√(3/4), which simplifies to cos x = ±√3 / 2. This is a crucial step, as it highlights the importance of considering all possible solutions when dealing with trigonometric equations. The ± sign tells us that cosine can be either positive or negative, which will lead to different angles within our interval. In the next section, we'll look at the angles where cosine takes these values and then consider the interval [π/4, 5π/4] to find our specific solutions.
Finding Possible Values of cos x
So, we've arrived at cos x = ±√3 / 2. This means we need to find the angles x where the cosine function equals either positive √3 / 2 or negative √3 / 2. Think back to the unit circle, guys! Cosine corresponds to the x-coordinate of a point on the unit circle. We know that cos(π/6) = √3 / 2. Therefore, π/6 is one reference angle we're interested in. However, we also need to consider where cosine is negative. Cosine is negative in the second and third quadrants. In the second quadrant, the angle with the same reference angle as π/6 is π - π/6 = 5π/6. And in the third quadrant, it's π + π/6 = 7π/6. So, we have three potential solutions so far: π/6, 5π/6, and 7π/6. But remember, we're only interested in solutions within the interval [π/4, 5π/4]. This is where we need to be careful and filter out any solutions that don't fall within this range. By understanding the unit circle and the behavior of the cosine function in different quadrants, we can systematically identify potential solutions. In the next step, we'll compare these potential solutions with our given interval and determine the final set of answers.
Considering the Interval [π/4, 5π/4]
Alright, let's put on our detective hats and narrow down the solutions. We have three potential values for x: π/6, 5π/6, and 7π/6. But remember our constraint: π/4 ≤ x ≤ 5π/4. This means we're only interested in angles that fall within this range. Let's compare each potential solution with the interval boundaries. First, π/6. Is π/6 greater than or equal to π/4? No, it's not. π/6 is approximately 0.52 radians, while π/4 is approximately 0.79 radians. So, π/6 is out. Next, let's consider 5π/6. Is 5π/6 within our interval? 5π/6 is approximately 2.62 radians. Since π/4 is approximately 0.79 radians and 5π/4 is approximately 3.93 radians, 5π/6 does indeed fall within our interval. So, 5π/6 is a valid solution. Finally, let's check 7π/6. 7π/6 is approximately 3.67 radians, which also falls within the interval [π/4, 5π/4]. Therefore, 7π/6 is also a valid solution. So, after carefully considering the interval, we've found two solutions that satisfy both the equation and the given constraint. This step is crucial because it ensures that our solutions are not just mathematically correct but also relevant to the specific problem we're trying to solve. In the final section, we'll summarize our findings and present the solutions.
Final Solution
We've reached the finish line, guys! After all the trigonometric maneuvering and algebraic gymnastics, we've successfully found the values of x that satisfy the equation 12cos²x - 3 = sec²x - tan²x + 5 within the interval [π/4, 5π/4]. Remember, we simplified the equation using trigonometric identities, solved for cos x, found potential angles using the unit circle, and then carefully considered the given interval to narrow down our solutions. Our final solutions are x = 5π/6 and x = 7π/6. These are the angles within the specified interval where the original equation holds true. This problem showcases the importance of understanding trigonometric identities, algebraic manipulation, and the unit circle in solving trigonometric equations. By mastering these skills, you'll be well-equipped to tackle a wide range of trigonometric challenges. Keep practicing, and you'll become a trig wizard in no time! Remember, the key to solving complex problems is to break them down into smaller, manageable steps and to use the tools and concepts you've learned along the way. Great job, everyone!
Therefore, the solutions are x = 5π/6 and x = 7π/6.