Solving Sin(x)tan²(x) = Sin(x) On [0, 2π)

by Alex Johnson 42 views

Hello there, math enthusiasts! Today, we're diving into the exciting world of trigonometry to solve a fascinating equation: sin(x)tan²(x) = sin(x). Our mission? To find all the solutions for x within the interval [0, 2π). This means we're looking for angles in radians that fall within one complete revolution of the unit circle. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's break down the equation and the interval we're working with. The equation sin(x)tan²(x) = sin(x) involves two trigonometric functions: sine (sin(x)) and tangent (tan(x)). Remember that tan(x) can also be expressed as sin(x)/cos(x). This is a crucial piece of information that we'll use later.

The interval [0, 2π) represents all angles from 0 radians up to, but not including, 2π radians. This is a full circle on the coordinate plane. When we find solutions, we need to make sure they fall within this range.

Key Concepts and Identities

To successfully solve this equation, we need to keep a few key trigonometric concepts and identities in mind:

  • The Unit Circle: Visualizing the unit circle is super helpful. It allows us to see the values of sine and cosine for different angles. Remember, sine corresponds to the y-coordinate, and cosine corresponds to the x-coordinate.
  • The Definition of Tangent: tan(x) = sin(x) / cos(x). This identity is essential for simplifying our equation.
  • Zero Product Property: If a * b = 0, then either a = 0 or b = 0 (or both). We'll use this to solve for x once we've factored the equation.
  • Periodicity of Trigonometric Functions: Sine and cosine functions are periodic, meaning their values repeat after a certain interval (2π for sine and cosine). Tangent is also periodic, but its period is π. We need to keep this in mind to make sure we find all solutions within our interval.

Step-by-Step Solution

Now, let's get our hands dirty and solve the equation step-by-step:

Step 1: Rearrange the Equation

The first thing we want to do is get all the terms on one side of the equation. This sets us up for factoring. Subtract sin(x) from both sides:

sin(x)tan²(x) - sin(x) = 0

Step 2: Factor out sin(x)

Notice that sin(x) is a common factor in both terms. Let's factor it out:

sin(x)(tan²(x) - 1) = 0

Step 3: Apply the Zero Product Property

Now we have a product of two factors that equals zero. This means that either sin(x) = 0 or (tan²(x) - 1) = 0.

Let's address each of these cases separately:

Case 1: sin(x) = 0

We need to find all the angles x in the interval [0, 2π) where the sine function is zero. Remember, sine corresponds to the y-coordinate on the unit circle. The y-coordinate is zero at two points: 0 radians and π radians.

So, our solutions from this case are:

  • x = 0
  • x = π

Case 2: tan²(x) - 1 = 0

Let's solve this equation for tan(x). First, add 1 to both sides:

tan²(x) = 1

Now, take the square root of both sides:

tan(x) = ±1

This gives us two separate equations to solve:

  • tan(x) = 1
  • tan(x) = -1
Solving tan(x) = 1

We need to find the angles x where the tangent function is 1. Remember that tangent is the ratio of sine to cosine (sin(x) / cos(x)). Tangent is 1 when sine and cosine have the same value. This occurs in the first and third quadrants.

Within our interval [0, 2π), the solutions are:

  • x = π/4 (in the first quadrant)
  • x = 5π/4 (in the third quadrant)
Solving tan(x) = -1

Now we need to find the angles x where the tangent function is -1. Tangent is -1 when sine and cosine have opposite signs. This occurs in the second and fourth quadrants.

Within our interval [0, 2π), the solutions are:

  • x = 3π/4 (in the second quadrant)
  • x = 7π/4 (in the fourth quadrant)

Step 4: Combine All Solutions

Now we need to gather all the solutions we found from each case. Our solutions for the equation sin(x)tan²(x) = sin(x) on the interval [0, 2π) are:

  • x = 0
  • x = π
  • x = π/4
  • x = 5π/4
  • x = 3π/4
  • x = 7π/4

Verification and Checking for Extraneous Solutions

It's always a good idea to verify our solutions, especially when dealing with trigonometric equations. We need to make sure that none of our solutions make the original equation undefined. In this case, we need to check for values that would make tan(x) undefined, which happens when cos(x) = 0.

Cosine is zero at π/2 and 3π/2. None of our solutions match these values, so we don't have any extraneous solutions!

Graphical Verification

Another way to verify our solutions is to graph the equation y = sin(x)tan²(x) and y = sin(x) on the same coordinate plane. The points where the graphs intersect represent the solutions to our equation. If you graph these functions, you'll see that they intersect at the x-values we found!

Tips and Tricks for Solving Trigonometric Equations

Solving trigonometric equations can be tricky, but here are some tips and tricks to make the process smoother:

  • Simplify: Use trigonometric identities to simplify the equation whenever possible. This often makes the equation easier to solve.
  • Factor: Factoring is a powerful tool for solving trigonometric equations. Look for common factors or patterns that allow you to factor the equation.
  • Isolate the Trigonometric Function: Try to isolate the trigonometric function (sin(x), cos(x), tan(x), etc.) on one side of the equation.
  • Use the Unit Circle: The unit circle is your best friend when solving trigonometric equations. Use it to visualize the values of sine, cosine, and tangent for different angles.
  • Check for Extraneous Solutions: Always check your solutions to make sure they don't make the original equation undefined.
  • General Solutions vs. Solutions in an Interval: Be mindful of whether you're asked to find all general solutions or just the solutions within a specific interval. General solutions involve adding multiples of the period to the solutions you find.
  • Practice, Practice, Practice: The more you practice, the better you'll become at solving trigonometric equations. Work through a variety of problems to build your skills.

Common Mistakes to Avoid

When solving trigonometric equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Dividing by a Trigonometric Function: Avoid dividing both sides of the equation by a trigonometric function (like sin(x) or cos(x)) because you might lose solutions. Instead, factor the equation and use the Zero Product Property.
  • Forgetting the ± Sign: When taking the square root of both sides of an equation, remember to include both the positive and negative square roots.
  • Ignoring the Interval: Make sure you're only including solutions that fall within the specified interval. If you're asked for solutions in [0, 2π), don't include solutions outside that range.
  • Not Checking for Extraneous Solutions: Always check your solutions to make sure they don't make the original equation undefined.
  • Confusing Radians and Degrees: Pay close attention to whether the problem is asking for solutions in radians or degrees, and make sure you're using the correct units.

Real-World Applications of Trigonometric Equations

You might be wondering, "When will I ever use this in the real world?" Well, trigonometric equations have numerous applications in various fields, including:

  • Physics: Modeling oscillations, waves, and projectile motion.
  • Engineering: Designing structures, analyzing circuits, and controlling systems.
  • Navigation: Calculating distances and directions.
  • Computer Graphics: Creating realistic images and animations.
  • Music: Analyzing sound waves and creating musical effects.

The principles we've discussed today can be applied to more complex trigonometric problems. Mastering these basics is crucial for anyone delving into advanced math, physics, or engineering. Keep practicing, and you'll find these concepts becoming second nature.

Conclusion

Congratulations! You've successfully navigated the world of trigonometric equations and found all the solutions for sin(x)tan²(x) = sin(x) on the interval [0, 2π). Remember, the key to solving these equations is to simplify, factor, use the unit circle, and check for extraneous solutions. Keep practicing, and you'll become a trigonometric equation-solving pro in no time!

If you're looking to further expand your understanding of trigonometric functions and equations, I highly recommend checking out resources like Khan Academy's Trigonometry Section. It’s an excellent platform for learning and reinforcing mathematical concepts.