Solving Trigonometric Equations: Find All Solutions

Hey guys! Ever get stuck trying to solve trigonometric equations? It can feel like navigating a maze sometimes, right? Well, don't worry, because in this guide, we're going to break down a specific problem step-by-step and learn some powerful techniques along the way. We'll be tackling the equation $3 \sin(4x) = -6 \sin(2x)$ and finding all its solutions within the interval $[0, 2\pi)$. So, buckle up, and let's dive in!

Understanding the Problem: Trigonometric Equations

Before we jump into solving, let's make sure we're all on the same page about what trigonometric equations actually are. In essence, these are equations that involve trigonometric functions like sine, cosine, tangent, and their reciprocals. Solving them means finding the values of the variable (in our case, 'x') that make the equation true. Trigonometric equations often have multiple solutions due to the periodic nature of trigonometric functions. This means their values repeat over regular intervals, which is why we often need to find solutions within a specific range, like the $[0, 2\pi)$ interval we're working with today.

In our specific equation, $3 \sin(4x) = -6 \sin(2x)$, we have sine functions with different arguments (4x and 2x). This adds a layer of complexity, as we can't directly combine the terms. We'll need to use trigonometric identities to simplify the equation and make it solvable. So, the main key here is understanding how trigonometric functions behave, what identities can help us, and how to apply those identities to find solutions systematically. Thinking about the unit circle and the graphs of sine and cosine can also give us valuable insights into potential solutions and the periodic nature of these functions. Recognizing these patterns is super helpful in solving these kinds of problems efficiently and accurately. We're basically detectives, using our trig knowledge to crack the case!

Step-by-Step Solution: Unraveling the Equation

Okay, let's get our hands dirty and actually solve this equation! We'll follow a step-by-step approach to make sure we don't miss anything.

1. Simplify the Equation

The first thing we want to do is simplify the equation as much as possible. We can start by dividing both sides of the equation by 3:

$\sin(4x) = -2 \sin(2x)$

This makes the numbers a little easier to deal with. Next, we need to address those different arguments (4x and 2x) in the sine functions. This is where our trigonometric identities come to the rescue!

2. Apply Trigonometric Identities

Remember the double-angle identity for sine? It states that:

$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$

We can use this identity to rewrite $\sin(4x)$ as $\sin(2(2x))$. Applying the identity, we get:

$\sin(4x) = 2 \sin(2x) \cos(2x)$

Now, substitute this back into our equation:

$2 \sin(2x) \cos(2x) = -2 \sin(2x)$

See how we're making progress? Both terms now have a $\sin(2x)$ factor, which we can use to further simplify the equation.

3. Rearrange and Factor

Let's move all the terms to one side to set the equation to zero:

$2 \sin(2x) \cos(2x) + 2 \sin(2x) = 0$

Now, we can factor out the common factor of $2 \sin(2x)$:

$2 \sin(2x) [\cos(2x) + 1] = 0$

This is a crucial step! We've transformed the equation into a product of factors equal to zero. This means that at least one of the factors must be zero for the equation to hold true.

4. Solve for Each Factor

Now we have two separate equations to solve:

• $2 \sin(2x) = 0$
• $\cos(2x) + 1 = 0$

Let's tackle them one at a time.

Solving $2 \sin(2x) = 0$

Divide both sides by 2:

$\sin(2x) = 0$

We need to find the values of $2x$ for which the sine function is zero. We know that $\sin(\theta) = 0$ when $\theta = n\pi$, where n is an integer. So:

$2x = n\pi$

Divide by 2:

$x = \frac{n\pi}{2}$

Now we need to find the values of x in the interval $[0, 2\pi)$. Let's plug in different integer values for 'n':

• n = 0: $x = 0$
• n = 1: $x = \frac{\pi}{2}$
• n = 2: $x = \pi$
• n = 3: $x = \frac{3\pi}{2}$
• n = 4: $x = 2\pi$ (but we exclude $2\pi$ since the interval is $[0, 2\pi)$)

So, the solutions from this factor are $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$.

Solving $\cos(2x) + 1 = 0$

Subtract 1 from both sides:

$\cos(2x) = -1$

We need to find the values of $2x$ for which the cosine function is -1. We know that $\cos(\theta) = -1$ when $\theta = (2n + 1)\pi$, where n is an integer. So:

$2x = (2n + 1)\pi$

Divide by 2:

$x = \frac{(2n + 1)\pi}{2}$

Now, let's find the values of x in the interval $[0, 2\pi)$:

• n = 0: $x = \frac{\pi}{2}$
• n = 1: $x = \frac{3\pi}{2}$
• n = 2: $x = \frac{5\pi}{2}$ (This is greater than $2\pi$, so we exclude it.)

So, the solutions from this factor are $x = \frac{\pi}{2}, \frac{3\pi}{2}$. Notice that these solutions are already included in the solutions we found from the first factor.

5. Combine the Solutions

Finally, we combine all the solutions we found:

$x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$

These are all the solutions to the equation $3 \sin(4x) = -6 \sin(2x)$ in the interval $[0, 2\pi)$.

Key Strategies for Solving Trigonometric Equations

Before we wrap up, let's highlight some key strategies that will help you tackle other trigonometric equations:

• Simplify: Always try to simplify the equation first by dividing or combining like terms.
• Use Identities: Trigonometric identities are your best friends! Learn them and know when to apply them. Double-angle, half-angle, Pythagorean, and sum-to-product identities are especially useful.
• Factor: Factoring can often break down a complex equation into simpler parts.
• Solve for Each Factor: When you have a product of factors equal to zero, solve each factor separately.
• Consider the Interval: Make sure your solutions fall within the specified interval. Add or subtract multiples of $2\pi$ (or the appropriate period for the function) to find solutions in the correct range.
• Check for Extraneous Solutions: Sometimes, operations like squaring both sides can introduce extraneous solutions, so always check your answers in the original equation.

Practice Makes Perfect: Your Next Steps

Solving trigonometric equations is like any other math skill – it gets easier with practice. Now that we've walked through this example together, it's your turn to try some on your own. Look for similar problems in your textbook or online, and don't be afraid to experiment with different approaches. If you get stuck, review the steps we covered in this guide, and remember those key strategies.

Also, visualizing the unit circle and the graphs of trigonometric functions can give you a much more intuitive understanding of the solutions. For example, you can quickly see where the sine function is zero or where the cosine function equals -1. This visual approach can be incredibly helpful for both solving and checking your answers.

So, go out there and conquer those trigonometric equations! With a little practice and the right strategies, you'll be solving them like a pro in no time. You've got this! Remember, the key is to understand the underlying principles and to approach each problem systematically. Good luck, and happy solving!