Solving 4sin^3(x) - 3sin(x) = 0: A Step-by-Step Guide

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Hey everyone! Today, we're diving into a trigonometric equation: 4sin^3(x) - 3sin(x) = 0. Don't worry, it might look intimidating at first, but we'll break it down step by step so it's super easy to understand. We'll explore the techniques and thought processes involved in tackling such problems. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's take a closer look at the equation: 4sin^3(x) - 3sin(x) = 0. The key here is recognizing that we have a cubic equation in terms of sin(x). This means we're dealing with a trigonometric identity in disguise! Specifically, this equation bears a striking resemblance to the triple angle identity for sine, which is sin(3x) = 3sin(x) - 4sin^3(x). Spotting these patterns is crucial in math, guys. It's like finding a secret key that unlocks the solution.

This equation is a classic example of how trigonometric identities can simplify complex expressions. By recognizing the relationship between the given equation and the sin(3x) identity, we can transform a seemingly complicated problem into a more manageable one. Think of trigonometric identities as the fundamental building blocks of trigonometric equations. They allow us to manipulate and rearrange expressions to reveal hidden structures and relationships. Mastering these identities is essential for anyone looking to excel in trigonometry. It's not just about memorizing formulas, but also about understanding how and when to apply them. It's like having a toolbox filled with different tools; knowing which tool to use for a specific task is key. For instance, the double-angle and half-angle identities are also powerful tools that can be used to simplify expressions and solve equations. They are derived from the basic trigonometric definitions and the Pythagorean identity, and they provide relationships between trigonometric functions of different angles. Understanding these relationships can significantly simplify problem-solving in trigonometry.

Step 1: Factoring out sin(x)

The first thing we can do is factor out sin(x) from the equation. This is a standard technique for solving polynomial-like equations. It helps us break down the problem into smaller, more manageable parts. So, let's do it: 4sin^3(x) - 3sin(x) = 0 becomes sin(x) [4sin^2(x) - 3] = 0. Now we have a product of two factors that equals zero. This means that at least one of the factors must be zero. This is a crucial step because it allows us to split the original equation into two simpler equations, each of which can be solved independently. Factoring is a fundamental algebraic technique that is used extensively in mathematics. It is a way of rewriting an expression as a product of simpler expressions, which can make it easier to analyze and solve. In this case, factoring out sin(x) allowed us to isolate the trigonometric function and simplify the remaining expression. This is a common strategy in solving trigonometric equations, as it can help to reduce the complexity of the problem and make it more accessible.

This gives us two possibilities:

  1. sin(x) = 0
  2. 4sin^2(x) - 3 = 0

Step 2: Solving sin(x) = 0

The first equation, sin(x) = 0, is straightforward. We need to find the values of x for which the sine function is zero. Think about the unit circle. Sine corresponds to the y-coordinate. So, where on the unit circle is the y-coordinate zero? At 0, π, 2π, and so on. In general, sin(x) = 0 when x = nπ, where n is an integer. These are the points where the unit circle intersects the x-axis.

Understanding the unit circle is crucial for solving trigonometric equations. The unit circle provides a visual representation of the trigonometric functions and their values at different angles. It allows us to easily identify the angles where sine, cosine, and tangent take on specific values. For instance, we can see that sin(x) = 0 at x = 0, π, 2π, and so on. This is because these angles correspond to points on the unit circle where the y-coordinate (which represents the sine value) is zero. Similarly, we can use the unit circle to find the angles where cosine is zero (at x = π/2, 3π/2, etc.) or where sine and cosine have specific values. The unit circle is an indispensable tool for anyone working with trigonometry.

So, the solutions for this part are x = nπ, where n is any integer (…, -2π, -π, 0, π, 2π, …).

Step 3: Solving 4sin^2(x) - 3 = 0

Now let's tackle the second equation: 4sin^2(x) - 3 = 0. This looks a bit more involved, but we'll handle it. First, we isolate sin^2(x):

4sin^2(x) = 3 sin^2(x) = 3/4

Now, we take the square root of both sides:

sin(x) = ±√(3/4) sin(x) = ±√3 / 2

So, we have two cases to consider: sin(x) = √3 / 2 and sin(x) = -√3 / 2. Remember guys, when taking the square root, we need to consider both positive and negative solutions.

Step 4: Solving sin(x) = √3 / 2

Let's solve sin(x) = √3 / 2. Again, think about the unit circle. Where is the y-coordinate equal to √3 / 2? This happens at x = π/3 and x = 2π/3 in the interval [0, 2π). But remember, the sine function is periodic with a period of 2π. This means that the solutions repeat every 2π radians. Therefore, the general solutions are x = π/3 + 2nπ and x = 2π/3 + 2nπ, where n is any integer.

Understanding the periodicity of trigonometric functions is crucial for finding all possible solutions to trigonometric equations. The sine and cosine functions have a period of 2π, which means that their values repeat every 2π radians. This is because the unit circle repeats itself every full rotation. The tangent function, on the other hand, has a period of π, which means its values repeat every π radians. When solving trigonometric equations, we need to consider all possible solutions within a given interval or find a general solution that represents all possible solutions. This is where the concept of periodicity comes into play. By adding multiples of the period to the initial solutions, we can generate all other solutions.

Step 5: Solving sin(x) = -√3 / 2

Now, let's solve sin(x) = -√3 / 2. On the unit circle, this occurs at x = 4π/3 and x = 5π/3 in the interval [0, 2π). Again, considering the periodicity, the general solutions are x = 4π/3 + 2nπ and x = 5π/3 + 2nπ, where n is any integer. These angles are located in the third and fourth quadrants, where the y-coordinate is negative.

Visualizing the unit circle is extremely helpful in identifying the angles that satisfy a given trigonometric equation. The unit circle provides a visual representation of the sine, cosine, and tangent functions, making it easier to determine their values at different angles. By understanding the relationships between the angles and the coordinates of the points on the unit circle, we can quickly find the solutions to trigonometric equations. For instance, when solving sin(x) = -√3 / 2, we can look for the points on the unit circle where the y-coordinate (which represents the sine value) is -√3 / 2. These points correspond to the angles 4π/3 and 5π/3, which are the solutions to the equation.

Step 6: Combining the Solutions

Finally, let's put all the solutions together. We have:

  • x = nÏ€
  • x = Ï€/3 + 2nÏ€
  • x = 2Ï€/3 + 2nÏ€
  • x = 4Ï€/3 + 2nÏ€
  • x = 5Ï€/3 + 2nÏ€

Where n is any integer. These are all the possible solutions to the original equation.

We can actually consolidate these solutions further by noticing a pattern. The solutions π/3, 2π/3, 4π/3, and 5π/3 are equally spaced around the unit circle at intervals of π/3. Therefore, we can express these solutions more compactly as x = π/3 + nπ and x = 2π/3 + nπ, where n is an integer. This compact representation captures all the solutions without redundancy.

Alternative Approach: Using the Triple Angle Identity

As we mentioned earlier, the equation 4sin^3(x) - 3sin(x) = 0 looks very similar to the triple angle identity for sine. Let's rewrite the identity to match our equation:

sin(3x) = 3sin(x) - 4sin^3(x)

Multiplying both sides by -1, we get:

-sin(3x) = 4sin^3(x) - 3sin(x)

So, our equation is equivalent to:

-sin(3x) = 0 sin(3x) = 0

Now, we need to find the values of 3x for which sine is zero. This occurs when 3x = nπ, where n is an integer. Dividing by 3, we get:

x = nπ/3

This single expression gives us all the solutions we found earlier! For example:

  • When n = 0, x = 0
  • When n = 1, x = Ï€/3
  • When n = 2, x = 2Ï€/3
  • When n = 3, x = Ï€
  • When n = 4, x = 4Ï€/3
  • When n = 5, x = 5Ï€/3
  • When n = 6, x = 2Ï€

And so on. This demonstrates the power of recognizing trigonometric identities. By using the triple angle identity, we were able to solve the equation much more efficiently. This alternative approach highlights the importance of having a solid understanding of trigonometric identities and being able to recognize patterns in equations. It's like having a shortcut that can save you a lot of time and effort.

Conclusion

So, guys, we've successfully solved the equation 4sin^3(x) - 3sin(x) = 0 using factoring and the unit circle, and then we confirmed our results using the triple angle identity. Remember, the key to tackling trigonometric equations is to break them down into smaller parts, use trigonometric identities wisely, and visualize the solutions on the unit circle. Keep practicing, and you'll become a pro at these in no time! Happy solving! We've explored different techniques and approaches, highlighting the importance of recognizing patterns, utilizing trigonometric identities, and understanding the unit circle. Trigonometric equations can seem daunting at first, but with practice and a solid understanding of the fundamentals, they can become much more manageable.