Solving 3sinx - 2cosx = 2 A Trigonometric Equation Guide

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Hey everyone! Let's dive into solving a trigonometric equation today. We're going to tackle the equation 3sinx - 2cosx = 2. Trigonometric equations can seem intimidating at first, but with the right approach, they become much more manageable. We'll break down the steps, explore the underlying concepts, and by the end, you'll feel confident in your ability to solve similar problems. So, buckle up and let's get started!

Understanding the Challenge

When we first glance at the equation 3sinx - 2cosx = 2, it's natural to wonder how to combine the sine and cosine terms. Unlike simple equations where we can isolate a single trigonometric function, this one involves both sine and cosine with different coefficients. This is where we need a clever strategy to simplify the equation and make it solvable. This type of equation is a linear trigonometric equation in sine and cosine, and there are a couple of common methods to solve it. We'll focus on the auxiliary angle method, which is a powerful technique for dealing with these kinds of problems. The key idea behind the auxiliary angle method is to transform the left-hand side of the equation into a single trigonometric function, either sine or cosine, with a phase shift. This transformation allows us to rewrite the equation in a simpler form that we can then solve using standard trigonometric techniques. Before we jump into the solution, let's briefly recap some essential trigonometric identities and concepts that will be helpful in our journey. Remember the Pythagorean identity? sin²x + cos²x = 1. This identity is a cornerstone of trigonometry and will often come in handy when manipulating trigonometric expressions. Also, recall the angle addition formulas for sine and cosine. These formulas allow us to express trigonometric functions of sums or differences of angles in terms of individual trigonometric functions. Specifically, we'll be using the formula for sine of a sum: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). By mastering these fundamental tools, you'll be well-equipped to solve a wide range of trigonometric problems. The auxiliary angle method is not just a trick; it's a technique rooted in these fundamental identities and transformations. It's about seeing the underlying structure of the equation and using our knowledge of trigonometry to simplify it. So, let's keep these concepts in mind as we move forward and see how they play a crucial role in solving our equation.

Method 1: The Auxiliary Angle Method

This method is a classic for solving equations of the form a sin x + b cos x = c. The core idea is to transform the left side into a single trigonometric function. Here's how we do it:

  1. Divide by the Magnitude:

    • Calculate the magnitude, R, of the coefficients: R = √(a² + b²). In our case, a = 3 and b = -2, so R = √(3² + (-2)²) = √13.
    • Divide both sides of the equation by R: (3/√13)sinx - (2/√13)cosx = 2/√13.

  2. Introduce the Auxiliary Angle:

    • We want to find an angle, let's call it φ, such that cos φ = 3/√13 and sin φ = 2/√13. Notice that these values satisfy the Pythagorean identity: (3/√13)² + (2/√13)² = 1. This is crucial because it allows us to interpret these values as cosine and sine of an angle.
    • We can find φ using the arctangent function: φ = arctan(2/3). This is the angle whose tangent is 2/3. We'll keep φ in this form for now.

  3. Rewrite the Equation:

    • Now we can rewrite our equation using the angle addition formula for sine. Recall that sin(x - φ) = sinx cos φ - cosx sin φ. Our equation now looks like this: sinx cos φ - cosx sin φ = 2/√13, which is equivalent to sin(x - φ) = 2/√13.

  4. Solve for x:

    • Let's introduce a new variable: y = x - φ. Our equation becomes sin y = 2/√13. Now we have a simple equation to solve for y.
    • The general solution for sin y = k (where |k| ≤ 1) is given by: y = arcsin(k) + 2Ï€n or y = Ï€ - arcsin(k) + 2Ï€n, where n is an integer.
    • In our case, y = arcsin(2/√13) + 2Ï€n or y = Ï€ - arcsin(2/√13) + 2Ï€n.

  5. Substitute Back:

    • Finally, we need to substitute back x - φ for y and solve for x: x = φ + arcsin(2/√13) + 2Ï€n or x = φ + Ï€ - arcsin(2/√13) + 2Ï€n.
    • Remember that φ = arctan(2/3). We can substitute this back into our solution to get the final answer.

This might seem like a lot of steps, but each one is logical and builds upon the previous one. The key takeaway is the transformation of the original equation into a simpler form using the auxiliary angle. This method is powerful because it allows us to handle equations with a combination of sine and cosine terms. By dividing by the magnitude and introducing the auxiliary angle, we effectively rotate the coordinate system in a way that aligns the equation with a single trigonometric function. This is a beautiful example of how mathematical transformations can simplify complex problems.

Method 2: Converting to Tangent of Half-Angle

Another approach involves using the tangent half-angle substitution. This method can be a bit more algebraic, but it's a valuable technique to have in your toolkit. Guys, sometimes, you just need another way to skin the cat, right? Here's how it works:

  1. The Tangent Half-Angle Substitution:

    • Let t = tan(x/2). We can express sinx and cosx in terms of t using the following identities:
      • sinx = (2t) / (1 + t²)
      • cosx = (1 - t²) / (1 + t²)

  2. Substitute into the Equation:

    • Substitute these expressions into our original equation 3sinx - 2cosx = 2:
      • 3 * (2t) / (1 + t²) - 2 * (1 - t²) / (1 + t²) = 2

  3. Simplify and Rearrange:

    • Multiply both sides by (1 + t²) to get rid of the denominators:
      • 6t - 2(1 - t²) = 2(1 + t²)
    • Expand and simplify:
      • 6t - 2 + 2t² = 2 + 2t²
      • 6t = 4

  4. Solve for t:

    • Divide both sides by 6:
      • t = 2/3

  5. Solve for x:

    • Since t = tan(x/2), we have tan(x/2) = 2/3.
    • Solve for x/2:
      • x/2 = arctan(2/3) + Ï€n, where n is an integer.
    • Multiply both sides by 2 to get x:
      • x = 2arctan(2/3) + 2Ï€n

  6. Check for Extraneous Solutions:

    • It's crucial to check for extraneous solutions when using the tangent half-angle substitution. This is because the substitution can introduce solutions that don't satisfy the original equation. In this case, we need to make sure that the solutions we found are valid by plugging them back into the original equation. However, in this particular problem, the solution we obtained is valid.

This method transforms the trigonometric equation into an algebraic equation, which can be easier to solve for some people. The key idea is to express sine and cosine in terms of the tangent of the half-angle. This substitution allows us to get rid of the trigonometric functions and work with a polynomial equation. However, it's crucial to remember to check for extraneous solutions, as this method can sometimes introduce them. So, while it might seem like a straightforward approach, you need to be careful and verify your solutions at the end. But hey, that's math for ya! Always keepin' us on our toes.

Comparing the Methods

Both the auxiliary angle method and the tangent half-angle substitution have their strengths and weaknesses. The auxiliary angle method is generally more elegant and avoids the potential for extraneous solutions, but it might involve dealing with more complicated trigonometric expressions. On the other hand, the tangent half-angle substitution transforms the equation into an algebraic one, which can be easier to solve, but it requires careful checking for extraneous solutions. Choosing the best method often comes down to personal preference and the specific equation you're dealing with. Sometimes, one method will be clearly easier than the other, while other times, it might be a matter of taste. The important thing is to be familiar with both techniques so you can choose the one that works best for you. Think of it like having two different tools in your toolbox. One might be better for a certain job, but it's always good to have options. And hey, the more tools you have, the more confident you'll feel tackling any trigonometric equation that comes your way. So, keep practicing and experimenting with different methods, and you'll become a trigonometric equation-solving pro in no time!

Final Thoughts and Key Takeaways

Solving trigonometric equations can be a rewarding experience. It's like piecing together a puzzle, where you use your knowledge of trigonometric identities and techniques to find the hidden solutions. The equation 3sinx - 2cosx = 2 is a great example of a problem that can be tackled using multiple approaches. We explored the auxiliary angle method, which transforms the equation into a simpler form by introducing an auxiliary angle, and the tangent half-angle substitution, which converts the equation into an algebraic one. Both methods demonstrate the power of mathematical transformations in simplifying complex problems.

The most important thing is to practice and develop your problem-solving skills. The more you work with trigonometric equations, the more comfortable you'll become with the various techniques and strategies. Don't be afraid to experiment with different approaches and see what works best for you. And remember, there's often more than one way to solve a problem. So, keep exploring, keep learning, and keep having fun with math! You've got this, guys!

Key takeaways from our discussion:

  • Equations of the form a sin x + b cos x = c can be solved using the auxiliary angle method or the tangent half-angle substitution.
  • The auxiliary angle method involves transforming the left side of the equation into a single trigonometric function.
  • The tangent half-angle substitution involves expressing sinx and cosx in terms of tan(x/2).
  • Always check for extraneous solutions when using the tangent half-angle substitution.
  • Practice is key to mastering trigonometric equation solving.