Solving 4 Cos X = -sin^2 X + 1 Find Solutions In [0, 2π)

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Hey guys! Let's dive into solving this trigonometric equation. It might look a bit intimidating at first, but don't worry, we'll break it down step by step. Our mission is to find all the values of x that satisfy the equation 4 cos x = -sin^2 x + 1 within the interval [0, 2π). This means we're looking for solutions between 0 and 2π radians, which is a full circle on the unit circle. So, buckle up, and let's get started!

Rewriting the Equation

In order to solve trigonometric equations, the very first thing we need to do is to get everything in terms of a single trigonometric function. Looking at our equation, 4 cos x = -sin^2 x + 1, we see both cosine and sine functions. The good news is, we can use a very famous identity to help us out here: the Pythagorean identity. You've probably heard of it – it's sin^2 x + cos^2 x = 1. This identity is the key to unlocking this problem! We can rearrange this identity to express sin^2 x in terms of cos^2 x. Subtracting sin^2 x from both sides, we get cos^2 x = 1 - sin^2 x. Now, if we rearrange it again, we get sin^2 x = 1 - cos^2 x. This is exactly what we need!

Now, let's substitute this expression for sin^2 x back into our original equation. We had 4 cos x = -sin^2 x + 1. Replacing sin^2 x with (1 - cos^2 x), we get: 4 cos x = -(1 - cos^2 x) + 1. See how we've eliminated the sine function and now we only have cosine? That's progress! Now, let's simplify this equation. Distributing the negative sign, we have 4 cos x = -1 + cos^2 x + 1. Notice that the -1 and +1 cancel each other out. This leaves us with 4 cos x = cos^2 x. We're getting closer to a form we can actually solve.

Now, to make things even clearer, let’s rearrange the equation so that all the terms are on one side. Subtracting 4 cos x from both sides, we get 0 = cos^2 x - 4 cos x. This is starting to look like a quadratic equation, which is something we know how to handle. We can rewrite this as cos^2 x - 4 cos x = 0. This form is much easier to work with. Now, we're ready for the next step: factoring!

Factoring and Solving for cos x

So, guys, look at the equation cos^2 x - 4 cos x = 0. Does it remind you of anything? It looks like a quadratic equation, right? Think of cos x as a single variable, say y. Then the equation becomes y^2 - 4y = 0. How do we solve this? By factoring! We can factor out a common factor of cos x from both terms. This gives us cos x (cos x - 4) = 0. This is a huge step forward because now we have a product of two factors that equals zero. Remember the zero product property? It says that if the product of two factors is zero, then at least one of the factors must be zero. So, either cos x = 0 or cos x - 4 = 0. Let's consider each case separately.

First, let's look at cos x = 0. This is a fundamental trigonometric equation. We need to find all the angles x in the interval [0, 2π) where the cosine function is equal to zero. Remember, cosine corresponds to the x-coordinate on the unit circle. So, we're looking for points on the unit circle where the x-coordinate is zero. These points occur at the top and bottom of the unit circle, which correspond to the angles π/2 and 3π/2. So, we have two potential solutions: x = π/2 and x = 3π/2.

Now, let's consider the second factor: cos x - 4 = 0. Adding 4 to both sides, we get cos x = 4. Now, this is where we need to pause and think critically. Remember that the cosine function is bounded between -1 and 1. In other words, the value of cos x can never be greater than 1 or less than -1. So, cos x = 4 has no solutions. There are no angles x for which the cosine is equal to 4. This is a crucial observation, and it saves us from chasing after nonexistent solutions.

So, we've found two potential solutions from the first factor and no solutions from the second factor. This means our solutions are x = π/2 and x = 3π/2. But, hold on! We're not quite done yet. We need to make sure these solutions actually satisfy the original equation. This is a crucial step called verification, and it helps us avoid extraneous solutions that might have crept in during our algebraic manipulations.

Verifying the Solutions

Okay, guys, we've found two potential solutions: x = π/2 and x = 3π/2. Now, we need to verify that these solutions actually work in the original equation, which is 4 cos x = -sin^2 x + 1. This step is important to make sure we haven't introduced any extraneous solutions along the way.

Let's start with x = π/2. We need to plug this value into the original equation and see if both sides are equal. So, we have 4 cos(π/2) = -sin^2(π/2) + 1. We know that cos(π/2) = 0 and sin(π/2) = 1. Substituting these values, we get 4 * 0 = -(1)^2 + 1, which simplifies to 0 = -1 + 1, and further simplifies to 0 = 0. Hooray! This is a true statement, so x = π/2 is indeed a solution.

Now, let's check x = 3π/2. Plugging this value into the original equation, we have 4 cos(3π/2) = -sin^2(3π/2) + 1. We know that cos(3π/2) = 0 and sin(3π/2) = -1. Substituting these values, we get 4 * 0 = -(-1)^2 + 1, which simplifies to 0 = -1 + 1, and further simplifies to 0 = 0. Awesome! This is also a true statement, so x = 3π/2 is also a solution.

So, we've verified that both of our potential solutions, x = π/2 and x = 3π/2, are actual solutions to the original equation. This means we've successfully found all the solutions in the interval [0, 2π). We're in the home stretch now!

Final Answer

Alright, guys! After all that work, we've finally arrived at the final answer. We set out to solve the equation 4 cos x = -sin^2 x + 1 in the interval [0, 2π). We rewrote the equation using the Pythagorean identity, factored it, solved for cos x, and verified our solutions. We found two solutions that satisfy the equation within the given interval. These solutions are x = π/2 and x = 3π/2. We need to express the answers in radians in terms of π, which we've already done!

So, there you have it! The solutions to the equation 4 cos x = -sin^2 x + 1 in the interval [0, 2π) are x = π/2, 3π/2. We solved it! Remember, the key to tackling trigonometric equations is to use identities to simplify them, factor when possible, and always verify your solutions. Great job, everyone!

x = π/2, 3π/2

Are you struggling with trigonometric equations? Do you need to find all the solutions within the interval [0, 2π)? You've come to the right place! In this comprehensive guide, we'll walk you through the process of solving the equation 4 cos x = -sin^2 x + 1, step by step. We'll cover everything from rewriting the equation using trigonometric identities to verifying your final solutions. So, whether you're a student tackling homework or just brushing up on your math skills, this article is for you. Let's dive in and conquer those trig equations!

Understanding Trigonometric Equations

Before we jump into solving the specific equation 4 cos x = -sin^2 x + 1, it's important to understand the basics of trigonometric equations. Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. Solving these equations means finding the values of the variable (usually x or θ) that make the equation true. However, unlike algebraic equations that often have a finite number of solutions, trigonometric equations can have infinitely many solutions due to the periodic nature of trigonometric functions. This is why we often specify an interval, such as [0, 2π), within which we want to find the solutions.

The interval [0, 2π) represents one full cycle of the trigonometric functions. In radians, 2π corresponds to 360 degrees, which is a complete circle. When we're asked to find solutions in the interval [0, 2π), we're essentially looking for all the angles within one full circle that satisfy the equation. This is where the unit circle comes in handy. The unit circle is a circle with a radius of 1 centered at the origin in the coordinate plane. It provides a visual representation of the values of sine and cosine for different angles.

Remember, the x-coordinate of a point on the unit circle corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. This connection between angles and coordinates is crucial for solving trigonometric equations. By understanding the unit circle, we can quickly identify the angles where sine and cosine have specific values. For instance, we know that cos(0) = 1, sin(0) = 0, cos(π/2) = 0, sin(π/2) = 1, and so on. These fundamental values are the building blocks for solving more complex trigonometric equations.

Another important concept is trigonometric identities. Trigonometric identities are equations that are true for all values of the variable. These identities are like tools in our toolbox – they allow us to rewrite and simplify trigonometric equations. The most fundamental identity is the Pythagorean identity, which we'll use extensively in this article. There are many other identities as well, such as the double-angle formulas, half-angle formulas, and sum-to-product formulas. Mastering these identities is essential for becoming a proficient trigonometric equation solver.

Step-by-Step Solution of 4 cos x = -sin^2 x + 1

Now that we've covered the basics, let's get down to business and solve the equation 4 cos x = -sin^2 x + 1. We'll break down the solution into manageable steps, explaining the reasoning behind each step. By following this process, you'll not only learn how to solve this specific equation but also develop a general strategy for tackling other trigonometric equations.

Step 1: Rewrite the Equation Using Trigonometric Identities

The first step in solving 4 cos x = -sin^2 x + 1 is to rewrite the equation in terms of a single trigonometric function. As we saw earlier, we have both cosine and sine functions present. To eliminate one of them, we'll use the Pythagorean identity: sin^2 x + cos^2 x = 1. We can rearrange this identity to express sin^2 x in terms of cos^2 x. Subtracting cos^2 x from both sides, we get sin^2 x = 1 - cos^2 x. This is exactly what we need!

Now, let's substitute this expression for sin^2 x back into our original equation. We have 4 cos x = -sin^2 x + 1. Replacing sin^2 x with (1 - cos^2 x), we get: 4 cos x = -(1 - cos^2 x) + 1. This substitution is a crucial step because it allows us to transform the equation into a form that only involves cosine. Now, we can simplify the equation further.

Distributing the negative sign, we have 4 cos x = -1 + cos^2 x + 1. The -1 and +1 cancel each other out, leaving us with 4 cos x = cos^2 x. This is a significant simplification. We've gone from an equation involving both sine and cosine to an equation involving only cosine. Now, let's rearrange the equation to get all the terms on one side. Subtracting 4 cos x from both sides, we get 0 = cos^2 x - 4 cos x. This can be rewritten as cos^2 x - 4 cos x = 0. This form resembles a quadratic equation, which is a key insight for the next step.

Step 2: Factor the Equation

Looking at the equation cos^2 x - 4 cos x = 0, we can see that it has a common factor of cos x in both terms. Factoring out cos x, we get cos x (cos x - 4) = 0. This is a crucial step because it transforms the equation into a product of two factors equal to zero. This is where the zero product property comes into play. The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means that either cos x = 0 or cos x - 4 = 0.

By factoring the equation, we've effectively split it into two simpler equations. Now, we can solve each equation separately. The first equation, cos x = 0, is a fundamental trigonometric equation that we can solve directly using our knowledge of the unit circle. The second equation, cos x - 4 = 0, requires a little more thought. Adding 4 to both sides, we get cos x = 4. However, we need to remember that the range of the cosine function is [-1, 1]. This means that the value of cos x can never be greater than 1 or less than -1. Therefore, the equation cos x = 4 has no solutions.

Step 3: Solve for x

Now that we've factored the equation and applied the zero product property, we have two potential equations to solve: cos x = 0 and cos x = 4. As we discussed in the previous step, cos x = 4 has no solutions because the cosine function is bounded between -1 and 1. So, we only need to focus on solving cos x = 0. To solve cos x = 0, we need to find all the angles x in the interval [0, 2π) where the cosine function is equal to zero. This is where our understanding of the unit circle becomes crucial.

Remember that the x-coordinate of a point on the unit circle corresponds to the cosine of the angle. So, we're looking for points on the unit circle where the x-coordinate is zero. These points occur at the top and bottom of the unit circle. The angle at the top of the unit circle is π/2 radians, and the angle at the bottom of the unit circle is 3π/2 radians. Therefore, the solutions to cos x = 0 in the interval [0, 2π) are x = π/2 and x = 3π/2. These are our potential solutions to the original equation.

Step 4: Verify the Solutions

We've found two potential solutions: x = π/2 and x = 3π/2. However, it's crucial to verify that these solutions actually work in the original equation, 4 cos x = -sin^2 x + 1. This step is important to avoid extraneous solutions, which are solutions that arise during the solving process but do not satisfy the original equation.

Let's start by verifying x = π/2. We substitute this value into the original equation: 4 cos(π/2) = -sin^2(π/2) + 1. We know that cos(π/2) = 0 and sin(π/2) = 1. Substituting these values, we get 4 * 0 = -(1)^2 + 1, which simplifies to 0 = -1 + 1, and further simplifies to 0 = 0. This is a true statement, so x = π/2 is indeed a solution.

Now, let's verify x = 3π/2. We substitute this value into the original equation: 4 cos(3π/2) = -sin^2(3π/2) + 1. We know that cos(3π/2) = 0 and sin(3π/2) = -1. Substituting these values, we get 4 * 0 = -(-1)^2 + 1, which simplifies to 0 = -1 + 1, and further simplifies to 0 = 0. This is also a true statement, so x = 3π/2 is also a solution.

Since both potential solutions satisfy the original equation, we can confidently conclude that the solutions are x = π/2 and x = 3π/2.

Key Takeaways for Solving Trigonometric Equations

Solving trigonometric equations can seem daunting at first, but by following a systematic approach, you can conquer even the most challenging problems. Here are some key takeaways to keep in mind:

  1. Rewrite the equation: Use trigonometric identities to express the equation in terms of a single trigonometric function. The Pythagorean identity is your best friend here.
  2. Factor the equation: If possible, factor the equation to simplify it. The zero product property is a powerful tool for solving factored equations.
  3. Solve for x: Use your knowledge of the unit circle and trigonometric values to find the solutions within the given interval.
  4. Verify the solutions: Always verify your solutions by substituting them back into the original equation to avoid extraneous solutions.

By mastering these steps and practicing regularly, you'll become a pro at solving trigonometric equations. So, keep practicing, and don't be afraid to tackle new challenges. You've got this!

Trigonometric equation, solve trigonometric equation, interval [0, 2π), cosine, sine, Pythagorean identity, unit circle, radians, verify solutions, trigonometric identities