Solving Trigonometric Equations Find Solutions For 2cos²x - Cosx = 1
Hey guys! Trigonometric equations can sometimes feel like a puzzle, but with the right approach, they become super manageable. Today, we’re going to dive deep into solving the equation 2cos²x - cosx = 1, focusing on finding all the solutions within the interval [0, 2π) or [0°, 360°]. Let's break it down step by step to make sure we nail this.
Understanding Trigonometric Equations
Before we jump into solving, let's quickly recap what trigonometric equations are all about. Trigonometric equations are essentially equations that involve trigonometric functions like sine, cosine, tangent, and their reciprocals. Solving them means finding the angles that satisfy the equation. These equations pop up in various fields, from physics and engineering to computer graphics and music theory. So, getting a good handle on them is really beneficial. When we're looking for solutions within a specific interval, like [0, 2π) or [0°, 360°], we’re essentially looking for all the angles within one full rotation of the unit circle that make the equation true.
Why This Interval Matters
The interval [0, 2π) (or [0°, 360°]) is crucial because it represents one complete cycle around the unit circle. Beyond this interval, the trigonometric functions repeat their values, meaning there are infinitely many solutions if we don't restrict our range. Focusing on this interval helps us find the principal solutions, which are the unique solutions within one cycle. These principal solutions then serve as the foundation for finding any other solutions if we were to consider a broader range. Think of it like finding the core set of answers that everything else builds upon. This is why it’s so important to master solving equations within this interval – it’s the key to understanding all possible solutions.
Step-by-Step Solution for 2cos²x - cosx = 1
Okay, let's get down to business and solve our equation: 2cos²x - cosx = 1. We'll take it one step at a time to keep things clear and easy to follow.
1. Rearrange the Equation
The first thing we want to do is rearrange the equation so that it equals zero. This sets us up nicely for factoring, which is a common technique for solving trigonometric equations. So, we subtract 1 from both sides of the equation:
2cos²x - cosx - 1 = 0
Now, our equation is in a quadratic-like form, which is a huge step forward.
2. Factor the Quadratic
Next up, we need to factor the quadratic expression. If it helps, you can think of cosx as a variable, say y, so the equation becomes 2y² - y - 1 = 0. This might make the factoring process a bit more familiar. We’re looking for two binomials that multiply to give us our quadratic expression. After some careful consideration, we find that:
(2cosx + 1)(cosx - 1) = 0
This is the completely factored form of the equation, and it's a crucial step in finding our solutions. Factoring allows us to break down a complex equation into simpler parts, making it much easier to solve.
3. Set Each Factor to Zero
Now comes the fun part! We use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means we set each factor equal to zero and solve for cosx:
- 2cosx + 1 = 0
- cosx - 1 = 0
This gives us two separate equations to solve, each leading to potential solutions for x.
4. Solve for cosx
Let's tackle each equation individually. For the first equation, 2cosx + 1 = 0, we subtract 1 from both sides and then divide by 2:
2cosx = -1
cosx = -1/2
For the second equation, cosx - 1 = 0, we simply add 1 to both sides:
cosx = 1
So now we have two simpler equations: cosx = -1/2 and cosx = 1. These are much easier to work with and will lead us to our solutions for x.
5. Find the Angles
Alright, we’re in the home stretch! Now we need to find the angles x within our interval [0, 2π) that satisfy cosx = -1/2 and cosx = 1. This is where our knowledge of the unit circle comes in handy.
For cosx = -1/2
We know that cosine represents the x-coordinate on the unit circle. So, we’re looking for angles where the x-coordinate is -1/2. This occurs in the second and third quadrants. The reference angle for cosx = 1/2 (ignoring the negative sign for now) is π/3. Therefore, the angles in the second and third quadrants where cosx = -1/2 are:
- x = π - π/3 = 2π/3
- x = π + π/3 = 4π/3
For cosx = 1
For cosx = 1, we’re looking for angles where the x-coordinate on the unit circle is 1. This occurs at 0 radians (or 0°). So,
- x = 0
6. List All Solutions
Finally, we compile all the solutions we've found within the interval [0, 2π). These are the angles that make our original equation 2cos²x - cosx = 1 true.
Our solutions are:
- x = 0
- x = 2π/3
- x = 4π/3
And that's it! We've successfully solved the trigonometric equation and found all the solutions within the specified interval.
Alternative Method: Using Degrees
For those of you who prefer working with degrees, let's quickly run through the same process using degrees instead of radians. This can sometimes feel more intuitive, especially if you're used to thinking in degrees.
1. Convert the Interval
First, we note that our interval in degrees is [0°, 360°], which is equivalent to [0, 2π) in radians. This represents one full rotation around the unit circle.
2. Solve for cosx (Same as Before)
We already went through the steps of rearranging the equation and factoring it, so we have the same two equations to solve for cosx:
- cosx = -1/2
- cosx = 1
3. Find the Angles in Degrees
Now, we need to find the angles in degrees that satisfy these equations.
For cosx = -1/2
We know that cosine is negative in the second and third quadrants. The reference angle for cosx = 1/2 is 60°. Therefore, the angles in the second and third quadrants where cosx = -1/2 are:
- x = 180° - 60° = 120°
- x = 180° + 60° = 240°
For cosx = 1
For cosx = 1, we’re looking for the angle where the x-coordinate on the unit circle is 1. This occurs at 0°.
- x = 0°
4. List All Solutions in Degrees
So, our solutions in degrees are:
- x = 0°
- x = 120°
- x = 240°
These solutions correspond directly to the radian solutions we found earlier, just expressed in a different unit.
Common Mistakes to Avoid
When solving trigonometric equations, it's easy to stumble upon some common pitfalls. Let's highlight a few key mistakes to watch out for so you can steer clear of them.
1. Forgetting the ± Sign
When taking the square root to solve for a trigonometric function, always remember to consider both the positive and negative roots. For example, if you have sin²x = 1/4, you need to take both sinx = 1/2 and sinx = -1/2 into account. Forgetting the negative root means missing out on half of the possible solutions.
2. Not Considering All Quadrants
Trigonometric functions can have the same value in multiple quadrants. For instance, sine is positive in both the first and second quadrants. Make sure to check all relevant quadrants when finding angles that satisfy your equation. A good way to visualize this is by using the unit circle and paying attention to the signs of the trigonometric functions in each quadrant.
3. Incorrectly Applying Identities
Trigonometric identities are powerful tools, but they need to be applied correctly. Double-check that you’re using the right identity for the situation and that you’re substituting values accurately. A small mistake in applying an identity can lead to completely wrong solutions.
4. Not Factoring Completely
Factoring is a common technique for solving trigonometric equations, but it’s crucial to factor completely. If you stop factoring prematurely, you might miss some solutions. Always look for common factors and continue factoring until you can’t factor any further.
5. Dividing by a Trigonometric Function
Avoid dividing both sides of an equation by a trigonometric function (like cosx or sinx) because you might lose solutions. Instead, move all terms to one side and factor out the trigonometric function. This ensures that you capture all possible solutions, including those where the function equals zero.
6. Not Checking for Extraneous Solutions
When you perform operations that can introduce extraneous solutions (like squaring both sides), it’s essential to check your final solutions in the original equation. This will help you identify and discard any solutions that don’t actually satisfy the equation.
7. Ignoring the Interval Restriction
If the problem specifies an interval for the solutions, make sure you only include solutions within that interval. Adding or subtracting multiples of 2π (or 360°) might give you solutions that are outside the specified range.
Practice Problems
To really nail solving trigonometric equations, practice is key! Here are a few problems you can try on your own. Work through them step by step, and don’t be afraid to refer back to the methods we discussed earlier. Remember, the more you practice, the more comfortable you’ll become with these types of problems.
- 2sin²x - sinx = 1
- cos²x - cosx = 0
- 2sin²x + 3sinx + 1 = 0
Try solving these within the interval [0, 2π) (or [0°, 360°]). Good luck, and have fun!
Conclusion
Solving the trigonometric equation 2cos²x - cosx = 1 involves a series of steps, from rearranging and factoring the equation to finding the angles on the unit circle. By following these steps carefully and avoiding common mistakes, you can confidently tackle similar problems. Remember, practice makes perfect, so keep working at it, and you’ll master these equations in no time! We covered solving both in radians and degrees, giving you a solid foundation no matter which unit you prefer. So, keep practicing, and you’ll become a trigonometric equation-solving pro in no time! Cheers, guys!