Solving Trigonometric Equations: A Step-by-Step Guide

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Hey guys! Let's dive into solving a trigonometric equation today. We're going to find all the solutions for the equation (csc⁡θ−12)(sec⁡θ−2)=0\left(\csc \theta-\frac{1}{2}\right)(\sec \theta-2)=0 within the interval [0,2π)[0, 2\pi). Don't worry, it might seem a bit daunting at first, but we'll break it down step by step. By the end of this, you'll be a pro at solving these kinds of problems! The key is to understand the fundamentals and apply them methodically. Let's get started!

Understanding the Problem: Core Concepts

Before we jump into the solution, let's make sure we're all on the same page regarding the core concepts involved. This problem requires us to understand the basic trigonometric functions, their reciprocals, and how to solve equations involving them. We are dealing with cosecant (csc) and secant (sec), which are reciprocals of sine and cosine, respectively. Remember these relationships because they're fundamental to solving this equation: csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} and sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}. Also, we're working within the interval [0,2π)[0, 2\pi), which means we need to find all the angles, measured in radians, that satisfy the equation and fall within this range. This interval represents one full rotation around the unit circle, so we need to consider all possible solutions within that rotation. The equation given to us is a product of two factors set equal to zero. A critical rule to remember is that if the product of two factors is zero, then at least one of the factors must be zero. This fact is the cornerstone of our solution strategy. Now that we have a clear understanding of the problem, we can move forward with confidence. We'll first focus on setting each factor to zero and solving those simpler equations. Then, we'll find the angles within our specified interval that meet those criteria. The trick is to be systematic and meticulous so that you do not miss any possible solutions. Ready to solve? Let's go!

Breaking Down the Trigonometric Equation

Alright, here we go! Our equation is (csc⁡θ−12)(sec⁡θ−2)=0\left(\csc \theta-\frac{1}{2}\right)(\sec \theta-2)=0. As we mentioned earlier, we know that for this equation to be true, either (csc⁡θ−12)=0(\csc \theta - \frac{1}{2}) = 0 or (sec⁡θ−2)=0(\sec \theta - 2) = 0 (or both!). We'll address these separately and then combine our findings at the end. Let's start with the first part: csc⁡θ−12=0\csc \theta - \frac{1}{2} = 0. To solve this, we'll isolate csc⁡θ\csc \theta which gives us csc⁡θ=12\csc \theta = \frac{1}{2}. However, remember that csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}. So, we can rewrite our equation as 1sin⁡θ=12\frac{1}{\sin \theta} = \frac{1}{2}. Now, if we cross-multiply (or take the reciprocal of both sides), we get sin⁡θ=2\sin \theta = 2. But wait a minute! We know that the sine function can only take on values between -1 and 1. Therefore, there are no solutions from the first part of the equation because sin⁡θ=2\sin \theta = 2 is impossible. Now, let's move on to the second part: sec⁡θ−2=0\sec \theta - 2 = 0. We can isolate sec⁡θ\sec \theta to get sec⁡θ=2\sec \theta = 2. Remembering that sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, we can rewrite this as 1cos⁡θ=2\frac{1}{\cos \theta} = 2. If we take the reciprocal of both sides, we get cos⁡θ=12\cos \theta = \frac{1}{2}. This looks promising! We're searching for angles where the cosine is equal to 1/2. In the next section, we'll identify these angles within our given interval. Keep going, you're doing great!

Solving for θ\theta: Finding the Angles

Now comes the exciting part – finding the angles! We found that we need to solve cos⁡θ=12\cos \theta = \frac{1}{2}. Think about the unit circle, the most important tool when dealing with trigonometric functions. Where does the cosine (the x-coordinate on the unit circle) equal 1/2? Remember the special angles, like π3\frac{\pi}{3}, π6\frac{\pi}{6}, etc. The cosine function equals 1/2 at π3\frac{\pi}{3} and 5π3\frac{5\pi}{3} within the interval [0,2π)[0, 2\pi). If you are unsure about these angles, grab a unit circle diagram or consider the cosine graph. Cosine is positive in the first and fourth quadrants. So, we will have one angle in the first quadrant and one in the fourth quadrant. For the first quadrant, the reference angle is π3\frac{\pi}{3}. In the fourth quadrant, the angle can be found by subtracting π3\frac{\pi}{3} from 2π2\pi. Mathematically, it is 2π−π3=6π3−π3=5π32\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}. Therefore, the solutions to cos⁡θ=12\cos \theta = \frac{1}{2} are θ=π3\theta = \frac{\pi}{3} and θ=5π3\theta = \frac{5\pi}{3}. These angles are within the interval [0,2π)[0, 2\pi), so they are valid solutions. These are the only two angles where the cosine equals one-half within a full rotation of the unit circle. Remember, understanding the unit circle and the properties of trigonometric functions is key here. Keep visualizing that unit circle! This visualization is your best friend when it comes to trigonometry!

Detailed Steps to Find the Solutions

Let's recap and formalize the steps we've taken to find the solutions:

  1. Identify the Equation: We started with (csc⁡θ−12)(sec⁡θ−2)=0\left(\csc \theta-\frac{1}{2}\right)(\sec \theta-2)=0.
  2. Break Down the Equation: We recognized that either csc⁡θ−12=0\csc \theta - \frac{1}{2} = 0 or sec⁡θ−2=0\sec \theta - 2 = 0.
  3. Solve the First Part: For csc⁡θ−12=0\csc \theta - \frac{1}{2} = 0, we found that sin⁡θ=2\sin \theta = 2, which has no solution since the range of sin⁡θ\sin \theta is [−1,1][-1, 1].
  4. Solve the Second Part: For sec⁡θ−2=0\sec \theta - 2 = 0, we found that cos⁡θ=12\cos \theta = \frac{1}{2}.
  5. Find the Angles: We identified that cos⁡θ=12\cos \theta = \frac{1}{2} at θ=π3\theta = \frac{\pi}{3} and θ=5π3\theta = \frac{5\pi}{3} within the interval [0,2π)[0, 2\pi).
  6. State the Solutions: The solutions are θ=π3,5π3\theta = \frac{\pi}{3}, \frac{5\pi}{3}.

By breaking down the problem into smaller, manageable steps and using our knowledge of trigonometric functions and the unit circle, we found the solution. It is very important to be methodical when solving trigonometric equations to avoid errors. Always double-check your solutions by plugging them back into the original equation to ensure they work!

Conclusion: Summarizing the Solutions

Alright, guys, we've made it! We successfully found all the solutions to the trigonometric equation (csc⁡θ−12)(sec⁡θ−2)=0\left(\csc \theta-\frac{1}{2}\right)(\sec \theta-2)=0 in the interval [0,2π)[0, 2\pi). The solutions are θ=π3\theta = \frac{\pi}{3} and θ=5π3\theta = \frac{5\pi}{3}. Remember, the most crucial things to remember are: (1) understanding the definitions of the trigonometric functions, especially cosecant and secant, (2) being familiar with the unit circle and special angles, and (3) breaking down complex equations into simpler parts. I strongly advise you to work through similar problems on your own to cement your understanding. Try varying the equations and intervals to challenge yourself. Keep practicing, and you'll get better at these problems. Good luck, and thanks for joining me today! Feel free to ask if you have any questions; I'm always happy to help. Keep up the great work and happy solving!